# -*- coding: utf-8 -*-
r"""
Elliptic curves over number fields

An elliptic curve `E` over a number field `K` can be given
by a Weierstrass equation whose coefficients lie in `K` or by
using ``base_extend`` on an elliptic curve defined over a subfield.

One major difference to elliptic curves over `\QQ` is that there
might not exist a global minimal equation over `K`, when `K` does
not have class number one.
Another difference is the lack of understanding of modularity for
general elliptic curves over general number fields.

Currently Sage can obtain local information about `E/K_v` for finite places
`v`, it has an interface to Denis Simon's script for 2-descent, it can compute
the torsion subgroup of the Mordell-Weil group `E(K)`, and it can work with
isogenies defined over `K`.

EXAMPLE::

    sage: K.<i> = NumberField(x^2+1)
    sage: E = EllipticCurve([0,4+i])
    sage: E.discriminant()
    -3456*i - 6480
    sage: P= E([i,2])
    sage: P+P
    (-2*i + 9/16 : -9/4*i - 101/64 : 1)

::

    sage: E.has_good_reduction(2+i)
    True
    sage: E.local_data(4+i)
    Local data at Fractional ideal (i + 4):
    Reduction type: bad additive
    Local minimal model: Elliptic Curve defined by y^2 = x^3 + (i+4) over Number Field in i with defining polynomial x^2 + 1
    Minimal discriminant valuation: 2
    Conductor exponent: 2
    Kodaira Symbol: II
    Tamagawa Number: 1
    sage: E.tamagawa_product_bsd()
    1

::

    sage: E.simon_two_descent()
    (1, 1, [(i : 2 : 1)])

::

    sage: E.torsion_order()
    1

::

    sage: E.isogenies_prime_degree(3)
    [Isogeny of degree 3 from Elliptic Curve defined by y^2 = x^3 + (i+4) over Number Field in i with defining polynomial x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (-27*i-108) over Number Field in i with defining polynomial x^2 + 1]

AUTHORS:

- Robert Bradshaw 2007

- John Cremona

- Chris Wuthrich

REFERENCE:

- [Sil] Silverman, Joseph H. The arithmetic of elliptic curves. Second edition. Graduate Texts in
  Mathematics, 106. Springer, 2009.

- [Sil2] Silverman, Joseph H. Advanced topics in the arithmetic of elliptic curves. Graduate Texts in
  Mathematics, 151. Springer, 1994.
"""

#*****************************************************************************
#       Copyright (C) 2007 Robert Bradshaw <robertwb@math.washington.edu>
#                          William Stein   <wstein@gmail.com>
#
#  Distributed under the terms of the GNU General Public License (GPL)
#
#    This code is distributed in the hope that it will be useful,
#    but WITHOUT ANY WARRANTY; without even the implied warranty of
#    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
#    General Public License for more details.
#
#  The full text of the GPL is available at:
#
#                  http://www.gnu.org/licenses/
#*****************************************************************************

from ell_field import EllipticCurve_field
import ell_point
import sage.matrix.all as matrix
from sage.rings.ring import Ring
from sage.rings.arith import gcd, prime_divisors
from sage.misc.all import prod
import ell_torsion
from ell_generic import is_EllipticCurve

from gp_simon import simon_two_descent
from constructor import EllipticCurve
from sage.rings.all import PolynomialRing, ZZ, QQ, RealField
import sage.misc.misc
from sage.misc.misc import verbose, forall
from sage.rings.integer import Integer
from sage.rings.arith import valuation

import gal_reps_number_field

class EllipticCurve_number_field(EllipticCurve_field):
    r"""
    Elliptic curve over a number field.

    EXAMPLES::

        sage: K.<i>=NumberField(x^2+1)
        sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35])
        Elliptic Curve defined by y^2 + i*x*y + (i+1)*y = x^3 + (i-1)*x^2 + (24*i+15)*x + (14*i+35) over Number Field in i with defining polynomial x^2 + 1
    """
    def __init__(self, K, ainvs):
        r"""
        EXAMPLES:

        A curve from the database of curves over `\QQ`, but over a larger field:

            sage: K.<i>=NumberField(x^2+1)
            sage: EllipticCurve(K,'389a1')
            Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-2)*x over Number Field in i with defining polynomial x^2 + 1

        Making the field of definition explicitly larger::

            sage: EllipticCurve(K,[0,-1,1,0,0])
            Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in i with defining polynomial x^2 + 1

        """
        self._known_points = []
        EllipticCurve_field.__init__(self, K, ainvs)

    _point = ell_point.EllipticCurvePoint_number_field

    def base_extend(self, R):
        """
        Return the base extension of ``self`` to `R`.

        EXAMPLES::

            sage: E = EllipticCurve('11a3')
            sage: K = QuadraticField(-5, 'a')
            sage: E.base_extend(K)
            Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in a with defining polynomial x^2 + 5

        Check that non-torsion points are remembered when extending
        the base field (see :trac:`16034`)::

            sage: E = EllipticCurve([1, 0, 1, -1751, -31352])
            sage: K.<d> = QuadraticField(5)
            sage: E.gens()
            [(52 : 111 : 1)]
            sage: EK = E.base_extend(K)
            sage: EK.gens()
            [(52 : 111 : 1)]

        """
        E = super(EllipticCurve_number_field, self).base_extend(R)
        if isinstance(E, EllipticCurve_number_field):
            E._known_points = [E([R(_) for _ in P.xy()]) for P in self._known_points if not P.is_zero()]
        return E

    def simon_two_descent(self, verbose=0, lim1=2, lim3=4, limtriv=2,
                          maxprob=20, limbigprime=30, known_points=None):
        r"""
        Return lower and upper bounds on the rank of the Mordell-Weil
        group `E(K)` and a list of points.

        This method is used internally by the :meth:`~rank`,
        :meth:`~rank_bounds` and :meth:`~gens` methods.

        INPUT:

        - ``self`` -- an elliptic curve `E` over a number field `K`

        - ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level

        - ``lim1`` -- (default: 2) limit on trivial points on quartics

        - ``lim3`` -- (default: 4) limit on points on ELS quartics

        - ``limtriv`` -- (default: 2) limit on trivial points on `E`

        - ``maxprob`` -- (default: 20)

        - ``limbigprime`` -- (default: 30) to distinguish between
          small and large prime numbers. Use probabilistic tests for
          large primes. If 0, don't use probabilistic tests.

        - ``known_points`` -- (default: None) list of known points on
          the curve

        OUTPUT: a triple ``(lower, upper, list)`` consisting of

        - ``lower`` (integer) -- lower bound on the rank

        - ``upper`` (integer) -- upper bound on the rank

        - ``list`` -- list of points in `E(K)`

        The integer ``upper`` is in fact an upper bound on the
        dimension of the 2-Selmer group, hence on the dimension of
        `E(K)/2E(K)`.  It is equal to the dimension of the 2-Selmer
        group except possibly if `E(K)[2]` has dimension 1.  In that
        case, ``upper`` may exceed the dimension of the 2-Selmer group
        by an even number, due to the fact that the algorithm does not
        perform a second descent.

        .. note::

           For non-quadratic number fields, this code does return, but
           it takes a long time.

        ALGORITHM:

        Uses Denis Simon's PARI/GP scripts from
        http://www.math.unicaen.fr/~simon/.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 23, 'a')
            sage: E = EllipticCurve(K, '37')
            sage: E == loads(dumps(E))
            True
            sage: E.simon_two_descent()
            (2, 2, [(0 : 0 : 1), (1/8*a + 5/8 : -3/16*a - 7/16 : 1)])
            sage: E.simon_two_descent(lim1=3, lim3=20, limtriv=5, maxprob=7, limbigprime=10)
            (2, 2, [(-1 : 0 : 1), (-1/8*a + 5/8 : -3/16*a - 9/16 : 1)])

        ::

            sage: K.<a> = NumberField(x^2 + 7, 'a')
            sage: E = EllipticCurve(K, [0,0,0,1,a]); E
            Elliptic Curve defined by y^2  = x^3 + x + a over Number Field in a with defining polynomial x^2 + 7

            sage: v = E.simon_two_descent(verbose=1); v
             elliptic curve: Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)
             Trivial points on the curve = [[1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
            #S(E/K)[2]    = 2
            #E(K)/2E(K)   = 2
            #III(E/K)[2]  = 1
            rank(E/K)     = 1
             listpoints = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
            (1, 1, [(1/2*a + 3/2 : -a - 2 : 1)])

            sage: v = E.simon_two_descent(verbose=2)
            K = bnfinit(y^2 + 7);
            a = Mod(y,K.pol);
            bnfellrank(K, [0, 0, 0, 1, a], [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]]);
             elliptic curve: Y^2 = x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)
              A = Mod(0, y^2 + 7)
              B = Mod(1, y^2 + 7)
              C = Mod(y, y^2 + 7)
            <BLANKLINE>
              Computing L(S,2)
              L(S,2) = [Mod(Mod(-1, y^2 + 7)*x^2 + Mod(-1/2*y + 1/2, y^2 + 7)*x + 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(-1, y^2 + 7)*x^2 + Mod(-1/2*y - 1/2, y^2 + 7)*x + 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(-1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(x^2 + 2, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x + Mod(1/2*y + 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x + Mod(1/2*y - 3/2, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
            <BLANKLINE>
              Computing the Selmer group
              #LS2gen = 2
               LS2gen = [Mod(Mod(-5, y^2 + 7)*x^2 + Mod(-3*y, y^2 + 7)*x + Mod(8, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7)), Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y - 1/2, y^2 + 7)*x - 1, x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))]
              Search for trivial points on the curve
             Trivial points on the curve = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)], [1, 1, 0], [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7), 1]]
              zc = Mod(Mod(-5, y^2 + 7)*x^2 + Mod(-3*y, y^2 + 7)*x + Mod(8, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
              Hilbert symbol (Mod(2, y^2 + 7),Mod(-5, y^2 + 7)) =
              zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y - 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
              Hilbert symbol (Mod(-2*y + 2, y^2 + 7),Mod(1, y^2 + 7)) =
              sol of quadratic equation = [1, 0, 1]~
              zc*z1^2 = Mod(Mod(2*y - 2, y^2 + 7)*x + Mod(2*y + 10, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
              quartic: (-1/2*y + 1/2)*Y^2 = x^4 + (-3*y - 15)*x^2 + (-8*y - 16)*x + (-11/2*y - 15/2)
              reduced: Y^2 = (-1/2*y + 1/2)*x^4 - 4*x^3 + (-3*y + 3)*x^2 + (2*y - 2)*x + (1/2*y + 3/2)
              not ELS at [2, [0, 1]~, 1, 1, [1, -2; 1, 0]]
              zc = Mod(Mod(1, y^2 + 7)*x^2 + Mod(1/2*y + 1/2, y^2 + 7)*x + Mod(-1, y^2 + 7), x^3 + Mod(1, y^2 + 7)*x + Mod(y, y^2 + 7))
              comes from the trivial point [Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]
              m1 = 1
              m2 = 1
            #S(E/K)[2]    = 2
            #E(K)/2E(K)   = 2
            #III(E/K)[2]  = 1
            rank(E/K)     = 1
             listpoints = [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]]
            v = [1, 1, [[Mod(1/2*y + 3/2, y^2 + 7), Mod(-y - 2, y^2 + 7)]]]
            sage: v
            (1, 1, [(1/2*a + 3/2 : -a - 2 : 1)])

        A curve with 2-torsion::

            sage: K.<a> = NumberField(x^2 + 7)
            sage: E = EllipticCurve(K, '15a')
            sage: E.simon_two_descent()  # long time (3s on sage.math, 2013), points can vary
            (1, 3, [...])

        Check that the bug reported in :trac:`15483` is fixed::

            sage: K.<s> = QuadraticField(229)
            sage: c4 = 2173 - 235*(1 - s)/2
            sage: c6 = -124369 + 15988*(1 - s)/2
            sage: E = EllipticCurve([-c4/48, -c6/864])
            sage: E.simon_two_descent()
            (0, 0, [])

            sage: R.<t> = QQ[]
            sage: L.<g> = NumberField(t^3 - 9*t^2 + 13*t - 4)
            sage: E1 = EllipticCurve(L,[1-g*(g-1),-g^2*(g-1),-g^2*(g-1),0,0])
            sage: E1.rank()  # long time (about 5 s)
            0

            sage: K = CyclotomicField(43).subfields(3)[0][0]
            sage: E = EllipticCurve(K, '37')
            sage: E.simon_two_descent()  # long time (4s on sage.math, 2013)
            (3, 3, [(0 : 0 : 1), (-1/4*zeta43_0^2 - 1/2*zeta43_0 + 3 : -3/8*zeta43_0^2 - 3/4*zeta43_0 + 4 : 1)])
        """
        verbose = int(verbose)
        if known_points is None:
            known_points = self._known_points
        known_points = [self(P) for P in known_points]

        # We deliberately do not use known_points as a key in the
        # following caching code, so that calling E.gens() a second
        # time (when known_points may have increased) will not cause
        # another execution of simon_two_descent.
        try:
            result = self._simon_two_descent_data[lim1,lim3,limtriv,maxprob,limbigprime]
            if verbose == 0:
                return result
        except AttributeError:
            self._simon_two_descent_data = {}
        except KeyError:
            pass

        t = simon_two_descent(self, verbose=verbose,
                              lim1=lim1, lim3=lim3, limtriv=limtriv,
                              maxprob=maxprob, limbigprime=limbigprime,
                              known_points=known_points)
        self._simon_two_descent_data[lim1,lim3,limtriv,maxprob,limbigprime] = t
        self._known_points.extend([P for P in t[2]
                                   if P not in self._known_points])
        return t

    def division_field(self, p, names, map=False, **kwds):
        """
        Given an elliptic curve over a number field `F` and a prime number `p`,
        construct the field `F(E[p])`.

        INPUT:

        - ``p`` -- a prime number (an element of `\ZZ`)

        - ``names`` -- a variable name for the number field

        - ``map`` -- (default: ``False``) also return an embedding of
          the :meth:`base_field` into the resulting field.

        - ``kwds`` -- additional keywords passed to
          :func:`sage.rings.number_field.splitting_field.splitting_field`.

        OUTPUT:

        If ``map`` is ``False``, the division field as an absolute number
        field.  If ``map`` is ``True``, a tuple ``(K, phi)`` where ``phi``
        is an embedding of the base field in the division field ``K``.

        .. WARNING::

            This takes a very long time when the degree of the division
            field is large (e.g. when `p` is large or when the Galois
            representation is surjective).  The ``simplify`` flag also
            has a big influence on the running time: sometimes
            ``simplify=False`` is faster, sometimes ``simplify=True``
            (the default) is faster.

        EXAMPLES:

        The 2-division field is the same as the splitting field of
        the 2-division polynomial (therefore, it has degree 1, 2, 3 or 6)::

            sage: E = EllipticCurve('15a1')
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x
            sage: E = EllipticCurve('14a1')
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x^2 + 5*x + 92
            sage: E = EllipticCurve('196b1')
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x^3 + x^2 - 114*x - 127
            sage: E = EllipticCurve('19a1')
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x^6 + 10*x^5 + 24*x^4 - 212*x^3 + 1364*x^2 + 24072*x + 104292

        For odd primes `p`, the division field is either the splitting
        field of the `p`-division polynomial, or a quadratic extension
        of it. ::

            sage: E = EllipticCurve('50a1')
            sage: F.<a> = E.division_polynomial(3).splitting_field(simplify_all=True); F
            Number Field in a with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3
            sage: K.<b> = E.division_field(3, simplify_all=True); K
            Number Field in b with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3

        If we take any quadratic twist, the splitting field of the
        3-division polynomial remains the same, but the 3-division field
        becomes a quadratic extension::

            sage: E = E.quadratic_twist(5)  # 50b3
            sage: F.<a> = E.division_polynomial(3).splitting_field(simplify_all=True); F
            Number Field in a with defining polynomial x^6 - 3*x^5 + 4*x^4 - 3*x^3 - 2*x^2 + 3*x + 3
            sage: K.<b> = E.division_field(3, simplify_all=True); K
            Number Field in b with defining polynomial x^12 - 3*x^11 + 8*x^10 - 15*x^9 + 30*x^8 - 63*x^7 + 109*x^6 - 144*x^5 + 150*x^4 - 120*x^3 + 68*x^2 - 24*x + 4

        Try another quadratic twist, this time over a subfield of `F`::

            sage: G.<c>,_,_ = F.subfields(3)[0]
            sage: E = E.base_extend(G).quadratic_twist(c); E
            Elliptic Curve defined by y^2 = x^3 + 5*a0*x^2 + (-200*a0^2)*x + (-42000*a0^2+42000*a0+126000) over Number Field in a0 with defining polynomial x^3 - 3*x^2 + 3*x + 9
            sage: K.<b> = E.division_field(3, simplify_all=True); K
            Number Field in b with defining polynomial x^12 - 10*x^10 + 55*x^8 - 60*x^6 + 75*x^4 + 1350*x^2 + 2025

        Some higher-degree examples::

            sage: E = EllipticCurve('11a1')
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x^6 + 2*x^5 - 48*x^4 - 436*x^3 + 1668*x^2 + 28792*x + 73844
            sage: K.<b> = E.division_field(3); K  # long time (3s on sage.math, 2014)
            Number Field in b with defining polynomial x^48 ...
            sage: K.<b> = E.division_field(5); K
            Number Field in b with defining polynomial x^4 - x^3 + x^2 - x + 1
            sage: E.division_field(5, 'b', simplify=False)
            Number Field in b with defining polynomial x^4 + x^3 + 11*x^2 + 41*x + 101
            sage: E.base_extend(K).torsion_subgroup()  # long time (2s on sage.math, 2014)
            Torsion Subgroup isomorphic to Z/5 + Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in b with defining polynomial x^4 - x^3 + x^2 - x + 1

            sage: E = EllipticCurve('27a1')
            sage: K.<b> = E.division_field(3); K
            Number Field in b with defining polynomial x^2 + 3*x + 9
            sage: K.<b> = E.division_field(2); K
            Number Field in b with defining polynomial x^6 + 6*x^5 + 24*x^4 - 52*x^3 - 228*x^2 + 744*x + 3844
            sage: K.<b> = E.division_field(2, simplify_all=True); K
            Number Field in b with defining polynomial x^6 - 3*x^5 + 5*x^3 - 3*x + 1
            sage: K.<b> = E.division_field(5); K   # long time (4s on sage.math, 2014)
            Number Field in b with defining polynomial x^48 ...
            sage: K.<b> = E.division_field(7); K  # long time (8s on sage.math, 2014)
            Number Field in b with defining polynomial x^72 ...

        Over a number field::

            sage: R.<x> = PolynomialRing(QQ)
            sage: K.<i> = NumberField(x^2 + 1)
            sage: E = EllipticCurve([0,0,0,0,i])
            sage: L.<b> = E.division_field(2); L
            Number Field in b with defining polynomial x^4 - x^2 + 1
            sage: L.<b>, phi = E.division_field(2, map=True); phi
            Ring morphism:
              From: Number Field in i with defining polynomial x^2 + 1
              To:   Number Field in b with defining polynomial x^4 - x^2 + 1
              Defn: i |--> -b^3
            sage: L.<b>, phi = E.division_field(3, map=True)
            sage: L
            Number Field in b with defining polynomial x^24 - 6*x^22 - 12*x^21 - 21*x^20 + 216*x^19 + 48*x^18 + 804*x^17 + 1194*x^16 - 13488*x^15 + 21222*x^14 + 44196*x^13 - 47977*x^12 - 102888*x^11 + 173424*x^10 - 172308*x^9 + 302046*x^8 + 252864*x^7 - 931182*x^6 + 180300*x^5 + 879567*x^4 - 415896*x^3 + 1941012*x^2 + 650220*x + 443089
            sage: phi
            Ring morphism:
              From: Number Field in i with defining polynomial x^2 + 1
              To:   Number Field in b with defining polynomial x^24 ...
              Defn: i |--> -215621657062634529/183360797284413355040732*b^23 ...

        AUTHORS:

        - Jeroen Demeyer (2014-01-06): :trac:`11905`, use
          ``splitting_field`` method, moved from ``gal_reps.py``, make
          it work over number fields.
        """
        p = Integer(p)
        if not p.is_prime():
            raise ValueError("p must be a prime number")

        verbose("Adjoining X-coordinates of %s-torsion points"%p)
        F = self.base_ring()
        f = self.division_polynomial(p)
        if p == 2:
            # For p = 2, the division field is the splitting field of
            # the division polynomial.
            return f.splitting_field(names, map=map, **kwds)

        # Compute splitting field of X-coordinates.
        # The Galois group of the division field is a subgroup of GL(2,p).
        # The Galois group of the X-coordinates is a subgroup of GL(2,p)/{-1,+1}.
        # We need the map to change the elliptic curve invariants to K.
        deg_mult = F.degree()*p*(p+1)*(p-1)*(p-1)//2
        K, F_to_K = f.splitting_field(names, degree_multiple=deg_mult, map=True, **kwds)

        verbose("Adjoining Y-coordinates of %s-torsion points"%p)

        # THEOREM (Cremona, http://trac.sagemath.org/ticket/11905#comment:21).
        # Let K be a field, E an elliptic curve over K and p an odd
        # prime number. Assume that K contains all roots of the
        # p-division polynomial of E. Then either K contains all
        # p-torsion points on E, or it doesn't contain any p-torsion
        # point.
        #
        # PROOF. Let G be the absolute Galois group of K (every element
        # in it fixes all elements of K). For any p-torsion point P
        # over the algebraic closure and any sigma in G, we must have
        # either sigma(P) = P or sigma(P) = -P (since K contains the
        # X-coordinate of P). Now assume that K does not contain all
        # p-torsion points. Then there exists a point P1 and a sigma in
        # G such that sigma(P1) = -P1. Now take a different p-torsion
        # point P2. Since sigma(P2) must be P2 or -P2 and
        # sigma(P1+P2) = sigma(P1)+sigma(P2) = sigma(P1)-P2 must
        # be P1+P2 or -(P1+P2), it follows that sigma(P2) = -sigma(P2).
        # Therefore, K cannot contain any p-torsion point.
        #
        # This implies that it suffices to adjoin the Y-coordinate
        # of just one point.

        # First factor f over F and then compute a root X of f over K.
        g = prime_divisors(f)[0]
        X = g.map_coefficients(F_to_K).roots(multiplicities=False)[0]

        # Polynomial defining the corresponding Y-coordinate
        a1,a2,a3,a4,a6 = (F_to_K(ai) for ai in self.a_invariants())
        rhs = X*(X*(X + a2) + a4) + a6
        RK = PolynomialRing(K, 'x')
        ypol = RK([-rhs, a1*X + a3, 1])
        L = ypol.splitting_field(names, map=map, **kwds)
        if map:
            L, K_to_L = L
            return L, F_to_K.post_compose(K_to_L)
        else:
            return L

    def height_pairing_matrix(self, points=None, precision=None):
        r"""
        Returns the height pairing matrix of the given points.

        INPUT:

        - points - either a list of points, which must be on this
          curve, or (default) None, in which case self.gens() will be
          used.

        - precision - number of bits of precision of result
          (default: None, for default RealField precision)

        EXAMPLES::

            sage: E = EllipticCurve([0, 0, 1, -1, 0])
            sage: E.height_pairing_matrix()
            [0.0511114082399688]

        For rank 0 curves, the result is a valid 0x0 matrix::

            sage: EllipticCurve('11a').height_pairing_matrix()
            []
            sage: E=EllipticCurve('5077a1')
            sage: E.height_pairing_matrix([E.lift_x(x) for x in [-2,-7/4,1]], precision=100)
            [  1.3685725053539301120518194471  -1.3095767070865761992624519454 -0.63486715783715592064475542573]
            [ -1.3095767070865761992624519454   2.7173593928122930896610589220   1.0998184305667292139777571432]
            [-0.63486715783715592064475542573   1.0998184305667292139777571432  0.66820516565192793503314205089]

            sage: E = EllipticCurve('389a1')
            sage: E = EllipticCurve('389a1')
            sage: P,Q = E.point([-1,1,1]),E.point([0,-1,1])
            sage: E.height_pairing_matrix([P,Q])
            [0.686667083305587 0.268478098806726]
            [0.268478098806726 0.327000773651605]

        Over a number field::

            sage: x = polygen(QQ)
            sage: K.<t> = NumberField(x^2+47)
            sage: EK = E.base_extend(K)
            sage: EK.height_pairing_matrix([EK(P),EK(Q)])
            [0.686667083305587 0.268478098806726]
            [0.268478098806726 0.327000773651605]

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,i,i])
            sage: P = E(-9+4*i,-18-25*i)
            sage: Q = E(i,-i)
            sage: E.height_pairing_matrix([P,Q])
            [  2.16941934493768 -0.870059380421505]
            [-0.870059380421505  0.424585837470709]
            sage: E.regulator_of_points([P,Q])
            0.164101403936070
        """
        if points is None:
            points = self.gens()
        else:
            for P in points:
                assert P.curve() == self

        r = len(points)
        if precision is None:
            RR = RealField()
        else:
            RR = RealField(precision)
        M = matrix.MatrixSpace(RR, r)
        mat = M()
        for j in range(r):
            mat[j,j] = points[j].height(precision=precision)
        for j in range(r):
            for k in range(j+1,r):
                mat[j,k]=((points[j]+points[k]).height(precision=precision) - mat[j,j] - mat[k,k])/2
                mat[k,j]=mat[j,k]
        return mat

    def regulator_of_points(self, points=[], precision=None):
        """
        Returns the regulator of the given points on this curve.

        INPUT:

        - ``points`` -(default: empty list)  a list of points on this curve

        - ``precision`` - int or None (default: None): the precision
          in bits of the result (default real precision if None)

        EXAMPLES::

            sage: E = EllipticCurve('37a1')
            sage: P = E(0,0)
            sage: Q = E(1,0)
            sage: E.regulator_of_points([P,Q])
            0.000000000000000
            sage: 2*P==Q
            True

        ::

            sage: E = EllipticCurve('5077a1')
            sage: points = [E.lift_x(x) for x in [-2,-7/4,1]]
            sage: E.regulator_of_points(points)
            0.417143558758384
            sage: E.regulator_of_points(points,precision=100)
            0.41714355875838396981711954462

        ::

            sage: E = EllipticCurve('389a')
            sage: E.regulator_of_points()
            1.00000000000000
            sage: points = [P,Q] = [E(-1,1),E(0,-1)]
            sage: E.regulator_of_points(points)
            0.152460177943144
            sage: E.regulator_of_points(points, precision=100)
            0.15246017794314375162432475705
            sage: E.regulator_of_points(points, precision=200)
            0.15246017794314375162432475704945582324372707748663081784028
            sage: E.regulator_of_points(points, precision=300)
            0.152460177943143751624324757049455823243727077486630817840280980046053225683562463604114816

        Examples over number fields::

            sage: K.<a> = QuadraticField(97)
            sage: E = EllipticCurve(K,[1,1])
            sage: P = E(0,1)
            sage: P.height()
            0.476223106404866
            sage: E.regulator_of_points([P])
            0.476223106404866

        ::

            sage: E = EllipticCurve('11a1')
            sage: x = polygen(QQ)
            sage: K.<t> = NumberField(x^2+47)
            sage: EK = E.base_extend(K)
            sage: T = EK(5,5)
            sage: T.order()
            5
            sage: P = EK(-2, -1/2*t - 1/2)
            sage: P.order()
            +Infinity
            sage: EK.regulator_of_points([P,T]) # random very small output
            -1.23259516440783e-32
            sage: EK.regulator_of_points([P,T]).abs() < 1e-30
            True

        ::

            sage: E = EllipticCurve('389a1')
            sage: P,Q = E.gens()
            sage: E.regulator_of_points([P,Q])
            0.152460177943144
            sage: K.<t> = NumberField(x^2+47)
            sage: EK = E.base_extend(K)
            sage: EK.regulator_of_points([EK(P),EK(Q)])
            0.152460177943144

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([0,0,0,i,i])
            sage: P = E(-9+4*i,-18-25*i)
            sage: Q = E(i,-i)
            sage: E.height_pairing_matrix([P,Q])
            [  2.16941934493768 -0.870059380421505]
            [-0.870059380421505  0.424585837470709]
            sage: E.regulator_of_points([P,Q])
            0.164101403936070

        """
        if points is None:
            points = []
        mat = self.height_pairing_matrix(points=points, precision=precision)
        return mat.det(algorithm="hessenberg")


    def is_local_integral_model(self,*P):
        r"""
        Tests if self is integral at the prime ideal `P`, or at all the
        primes if `P` is a list or tuple.

        INPUT:

        - ``*P`` -- a prime ideal, or a list or tuple of primes.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: P1,P2 = K.primes_above(5)
            sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
            sage: E.is_local_integral_model(P1,P2)
            False
            sage: Emin = E.local_integral_model(P1,P2)
            sage: Emin.is_local_integral_model(P1,P2)
            True
        """
        if len(P)==1: P=P[0]
        if isinstance(P,(tuple,list)):
            return forall(P, lambda x : self.is_local_integral_model(x))[0]
        return forall(self.ainvs(), lambda x : x.valuation(P) >= 0)[0]

    def local_integral_model(self,*P):
        r"""
        Return a model of self which is integral at the prime ideal
        `P`.

        .. note::

           The integrality at other primes is not affected, even if
           `P` is non-principal.

        INPUT:

        - ``*P`` -- a prime ideal, or a list or tuple of primes.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: P1,P2 = K.primes_above(5)
            sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
            sage: E.local_integral_model((P1,P2))
            Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1
        """
        if len(P)==1: P=P[0]
        if isinstance(P,(tuple,list)):
            E=self
            for Pi in P: E=E.local_integral_model(Pi)
            return E
        ai = self.a_invariants()
        e  = min([(ai[i].valuation(P)/[1,2,3,4,6][i]) for i in range(5)]).floor()
        pi = self.base_field().uniformizer(P, 'negative')
        return EllipticCurve([ai[i]/pi**(e*[1,2,3,4,6][i]) for i in range(5)])

    def is_global_integral_model(self):
        r"""
        Return true iff self is integral at all primes.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
            sage: P1,P2 = K.primes_above(5)
            sage: Emin = E.global_integral_model()
            sage: Emin.is_global_integral_model()
            True
        """
        return forall(self.a_invariants(), lambda x : x.is_integral())[0]

    def global_integral_model(self):
        r"""
        Return a model of self which is integral at all primes.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: E = EllipticCurve([i/5,i/5,i/5,i/5,i/5])
            sage: P1,P2 = K.primes_above(5)
            sage: E.global_integral_model()
            Elliptic Curve defined by y^2 + (-i)*x*y + (-25*i)*y = x^3 + 5*i*x^2 + 125*i*x + 3125*i over Number Field in i with defining polynomial x^2 + 1

        :trac:`7935`::

            sage: K.<a> = NumberField(x^2-38)
            sage: E = EllipticCurve([a,1/2])
            sage: E.global_integral_model()
            Elliptic Curve defined by y^2 = x^3 + 1444*a*x + 27436 over Number Field in a with defining polynomial x^2 - 38

        :trac:`9266`::

            sage: K.<s> = NumberField(x^2-5)
            sage: w = (1+s)/2
            sage: E = EllipticCurve(K,[2,w])
            sage: E.global_integral_model()
            Elliptic Curve defined by y^2 = x^3 + 2*x + (1/2*s+1/2) over Number Field in s with defining polynomial x^2 - 5

        :trac:`12151`::

            sage: K.<v> = NumberField(x^2 + 161*x - 150)
            sage: E = EllipticCurve([25105/216*v - 3839/36, 634768555/7776*v - 98002625/1296, 634768555/7776*v - 98002625/1296, 0, 0])
            sage: E.global_integral_model()
            Elliptic Curve defined by y^2 + (2094779518028859*v-1940492905300351)*x*y + (477997268472544193101178234454165304071127500*v-442791377441346852919930773849502871958097500)*y = x^3 + (26519784690047674853185542622500*v-24566525306469707225840460652500)*x^2 over Number Field in v with defining polynomial x^2 + 161*x - 150

        :trac:`14476`::

            sage: R.<t> = QQ[]
            sage: K.<g> = NumberField(t^4 - t^3 - 3*t^2 - t + 1)
            sage: E = EllipticCurve([ -43/625*g^3 + 14/625*g^2 - 4/625*g + 706/625, -4862/78125*g^3 - 4074/78125*g^2 - 711/78125*g + 10304/78125,  -4862/78125*g^3 - 4074/78125*g^2 - 711/78125*g + 10304/78125, 0,0])
            sage: E.global_integral_model()
            Elliptic Curve defined by y^2 + (15*g^3-48*g-42)*x*y + (-111510*g^3-162162*g^2-44145*g+37638)*y = x^3 + (-954*g^3-1134*g^2+81*g+576)*x^2 over Number Field in g with defining polynomial t^4 - t^3 - 3*t^2 - t + 1

        """
        K = self.base_field()
        ai = self.a_invariants()
        Ps = [[ ff[0] for ff in a.denominator_ideal().factor() ] for a in ai if not a.is_integral() ]
        Ps = sage.misc.misc.union(sage.misc.flatten.flatten(Ps))
        for P in Ps:
            pi = K.uniformizer(P,'positive')
            e  = min([(ai[i].valuation(P)/[1,2,3,4,6][i]) for i in range(5)]).floor()
            if e < 0 :
                ai = [ai[i]/pi**(e*[1,2,3,4,6][i]) for i in range(5)]
            if all(a.is_integral() for a in ai):
                break
        for z in ai:
            assert z.is_integral(), "bug in global_integral_model: %s" % list(ai)
        return EllipticCurve(list(ai))

    integral_model = global_integral_model

    def _reduce_model(self):
        r"""
        Returns a reduced model for this elliptic curve.

        Transforms the elliptic curve to a model which is optimally scaled
        with respect to units and in which `a_1`, `a_2`, `a_3` are
        reduced modulo 2, 3, 2 respectively.

        .. note::

           This only works on integral models, i.e. it requires that
           `a_1`, `a_2` and `a_3` lie in the ring of integers of the base
           field.

        EXAMPLES::

            sage: K.<a>=NumberField(x^2-38)
            sage: E=EllipticCurve([a, -5*a + 19, -39*a + 237, 368258520200522046806318224*a - 2270097978636731786720858047, 8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505])
            sage: E.ainvs()
            (a,
            -5*a + 19,
            -39*a + 237,
            368258520200522046806318224*a - 2270097978636731786720858047,
            8456608930180227786550494643437985949781*a - 52130038506835491453281450568107193773505)
            sage: E._reduce_model().ainvs()
            (a,
            a + 1,
            a + 1,
            368258520200522046806318444*a - 2270097978636731786720859345,
            8456608930173478039472018047583706316424*a - 52130038506793883217874390501829588391299)
            sage: EllipticCurve([101,202,303,404,505])._reduce_model().ainvs()
            (1, 1, 0, -2509254, 1528863051)
            sage: EllipticCurve([-101,-202,-303,-404,-505])._reduce_model().ainvs()
            (1, -1, 0, -1823195, 947995262)

            sage: E = EllipticCurve([a/4, 1])
            sage: E._reduce_model()
            Traceback (most recent call last):
            ...
            TypeError: _reduce_model() requires an integral model.
        """
        K = self.base_ring()
        ZK = K.maximal_order()
        try:
            (a1, a2, a3, a4, a6) = [ZK(a) for a in self.a_invariants()]
        except TypeError:
            import sys
            raise TypeError, "_reduce_model() requires an integral model.", sys.exc_info()[2]

        # N.B. Must define s, r, t in the right order.
        if ZK.absolute_degree() == 1:
            s = ((-a1)/2).round('up')
            r = ((-a2 + s*a1 +s*s)/3).round()
            t = ((-a3 - r*a1)/2).round('up')
        else:
            pariK = K._pari_()
            s = K(pariK.nfeltdiveuc(-a1, 2))
            r = K(pariK.nfeltdiveuc(-a2 + s*a1 + s*s, 3))
            t = K(pariK.nfeltdiveuc(-a3 - r*a1, 2))

        return self.rst_transform(r, s, t)

    def _scale_by_units(self):
        r""" Return a model reduced with respect to scaling by units.

        OUTPUT:

        A model for this elliptic curve, optimally scaled with repect
        to scaling by units when the base field is real quadratic;
        unchanged otherwise.

        .. note::

           This is currently only implemented for real quadratic fields.

        EXAMPLE::

           sage: K.<a> = NumberField(x^2-10)
           sage: u = K.units()[0]
           sage: E = EllipticCurve([0, 0, 0, 4536*a + 14148, -163728*a - 474336])
           sage: E1 = E.scale_curve(u^5)
           sage: E1.ainvs()
           (0,
           0,
           0,
           28087920796764302856*a + 88821804456186580548,
           -77225139016967233228487820912*a - 244207331916752959911655344864)
           sage: E1._scale_by_units().ainvs()
           (0, 0, 0, 4536*a + 14148, -163728*a - 474336)
        """
        K = self.base_field()
        if not K.signature()==(2,0):
            return self

        R = RealField(1000)  # lower precision works badly!
        embs = K.embeddings(R)
        u = K.units()[0]
        uv = [e(u).abs().log() for e in embs]

        c4, c6 = self.c_invariants()
        c4s = [e(c4) for e in embs]
        c6s = [e(c6) for e in embs]
        v = [(x4.abs()**(R.one()/4)+x6.abs()**(R.one()/6)).log() for x4,x6 in zip(c4s,c6s)]
        kr = -(v[0]*uv[0]+v[1]*uv[1])/(uv[0]**2+uv[1]**2)
        k1 = kr.floor()
        k2 = kr.ceil()
        nv1 = (v[0] + k1*uv[0])**2 + (v[1] + k1*uv[1])**2
        nv2 = (v[0] + k2*uv[0])**2 + (v[1] + k2*uv[1])**2
        if nv1 < nv2:
            k=k1
        else:
            k=k2
        return self.scale_curve(u**k)



    def local_data(self, P=None, proof=None, algorithm="pari", globally=False):
        r"""
        Local data for this elliptic curve at the prime `P`.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        - ``algorithm`` (string, default: "pari") -- Ignored unless the
          base field is `\QQ`.  If "pari", use the PARI C-library
          ``ellglobalred`` implementation of Tate's algorithm over
          `\QQ`. If "generic", use the general number field
          implementation.

        - ``globally`` -- whether the local algorithm uses global generators
          for the prime ideals. Default is False, which won't require any
          information about the class group. If True, a generator for `P`
          will be used if `P` is principal. Otherwise, or if ``globally``
          is False, the minimal model returned will preserve integrality
          at other primes, but not minimality.

        OUTPUT:

        If `P` is specified, returns the ``EllipticCurveLocalData``
        object associated to the prime `P` for this curve.  Otherwise,
        returns a list of such objects, one for each prime `P` in the
        support of the discriminant of this model.

        .. note::

           The model is not required to be integral on input.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: E = EllipticCurve([1 + i, 0, 1, 0, 0])
            sage: E.local_data()
            [Local data at Fractional ideal (2*i + 1):
            Reduction type: bad non-split multiplicative
            Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
            Minimal discriminant valuation: 1
            Conductor exponent: 1
            Kodaira Symbol: I1
            Tamagawa Number: 1,
            Local data at Fractional ideal (-3*i - 2):
            Reduction type: bad split multiplicative
            Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
            Minimal discriminant valuation: 2
            Conductor exponent: 1
            Kodaira Symbol: I2
            Tamagawa Number: 2]
            sage: E.local_data(K.ideal(3))
            Local data at Fractional ideal (3):
            Reduction type: good
            Local minimal model: Elliptic Curve defined by y^2 + (i+1)*x*y + y = x^3 over Number Field in i with defining polynomial x^2 + 1
            Minimal discriminant valuation: 0
            Conductor exponent: 0
            Kodaira Symbol: I0
            Tamagawa Number: 1

        An example raised in :trac:`3897`::

            sage: E = EllipticCurve([1,1])
            sage: E.local_data(3)
            Local data at Principal ideal (3) of Integer Ring:
            Reduction type: good
            Local minimal model: Elliptic Curve defined by y^2 = x^3 + x + 1 over Rational Field
            Minimal discriminant valuation: 0
            Conductor exponent: 0
            Kodaira Symbol: I0
            Tamagawa Number: 1
        """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        if P is None:
            primes = self.base_ring()(self.integral_model().discriminant()).support()
            return [self._get_local_data(pr, proof) for pr in primes]

        from sage.schemes.elliptic_curves.ell_local_data import check_prime
        P = check_prime(self.base_field(),P)

        return self._get_local_data(P,proof,algorithm,globally)

    def _get_local_data(self, P, proof, algorithm="pari", globally=False):
        r"""
        Internal function to create data for this elliptic curve at the prime `P`.

        This function handles the caching of local data.  It is called
        by local_data() which is the user interface and which parses
        the input parameters `P` and proof.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        - ``algorithm`` (string, default: "pari") -- Ignored unless the
          base field is `\QQ`.  If "pari", use the PARI C-library
          ``ellglobalred`` implementation of Tate's algorithm over
          `\QQ`. If "generic", use the general number field
          implementation.

        - ``globally`` -- whether the local algorithm uses global generators
          for the prime ideals. Default is False, which won't require any
          information about the class group. If True, a generator for `P`
          will be used if `P` is principal. Otherwise, or if ``globally``
          is False, the minimal model returned will preserve integrality
          at other primes, but not minimality.

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: E = EllipticCurve(K,[0,1,0,-160,308])
            sage: p = K.ideal(i+1)
            sage: E._get_local_data(p, False)
            Local data at Fractional ideal (i + 1):
            Reduction type: good
            Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in i with defining polynomial x^2 + 1
            Minimal discriminant valuation: 0
            Conductor exponent: 0
            Kodaira Symbol: I0
            Tamagawa Number: 1

        Verify that we cache based on the proof value and the algorithm choice::

            sage: E._get_local_data(p, False) is E._get_local_data(p, True)
            False

            sage: E._get_local_data(p, None, "pari") is E._get_local_data(p, None, "generic")
            False
        """
        try:
            return self._local_data[P, proof, algorithm, globally]
        except AttributeError:
            self._local_data = {}
        except KeyError:
            pass
        from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
        self._local_data[P, proof, algorithm, globally] = EllipticCurveLocalData(self, P, proof, algorithm, globally)
        return self._local_data[P, proof, algorithm, globally]

    def local_minimal_model(self, P, proof = None, algorithm="pari"):
        r"""
        Returns a model which is integral at all primes and minimal at `P`.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        - ``algorithm`` (string, default: "pari") -- Ignored unless the
          base field is `\QQ`.  If "pari", use the PARI C-library
          ``ellglobalred`` implementation of Tate's algorithm over
          `\QQ`. If "generic", use the general number field
          implementation.

        OUTPUT:

        A model of the curve which is minimal (and integral) at `P`.

        .. note::

           The model is not required to be integral on input.

           For principal `P`, a generator is used as a uniformizer,
           and integrality or minimality at other primes is not
           affected.  For non-principal `P`, the minimal model
           returned will preserve integrality at other primes, but not
           minimality.

        EXAMPLES::

            sage: K.<a>=NumberField(x^2-5)
            sage: E=EllipticCurve([20, 225, 750, 1250*a + 6250, 62500*a + 15625])
            sage: P=K.ideal(a)
            sage: E.local_minimal_model(P).ainvs()
            (0, 1, 0, 2*a - 34, -4*a + 66)
        """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        return self.local_data(P, proof, algorithm).minimal_model()

    def has_good_reduction(self, P):
        r"""
        Return True if this elliptic curve has good reduction at the prime `P`.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
          element generating such an ideal.

        OUTPUT:

        (bool) -- True if the curve has good reduction at `P`, else False.

        .. note::

           This requires determining a local integral minimal model;
           we do not just check that the discriminant of the current
           model has valuation zero.

        EXAMPLES::

            sage: E=EllipticCurve('14a1')
            sage: [(p,E.has_good_reduction(p)) for p in prime_range(15)]
            [(2, False), (3, True), (5, True), (7, False), (11, True), (13, True)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_good_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), True),
            (Fractional ideal (2*a + 1), False)]
        """
        return self.local_data(P).has_good_reduction()

    def has_bad_reduction(self, P):
        r"""
        Return True if this elliptic curve has bad reduction at the prime `P`.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
          element generating such an ideal.

        OUTPUT:

        (bool) True if the curve has bad reduction at `P`, else False.

        .. note::

           This requires determining a local integral minimal model;
           we do not just check that the discriminant of the current
           model has valuation zero.

        EXAMPLES::

            sage: E=EllipticCurve('14a1')
            sage: [(p,E.has_bad_reduction(p)) for p in prime_range(15)]
            [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_bad_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), False),
            (Fractional ideal (2*a + 1), True)]
        """
        return self.local_data(P).has_bad_reduction()

    def has_multiplicative_reduction(self, P):
        r"""
        Return True if this elliptic curve has (bad) multiplicative reduction at the prime `P`.

        .. note::

           See also ``has_split_multiplicative_reduction()`` and
           ``has_nonsplit_multiplicative_reduction()``.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
                 element generating such an ideal.

        OUTPUT:

        (bool) True if the curve has multiplicative reduction at `P`,
        else False.

        EXAMPLES::

            sage: E=EllipticCurve('14a1')
            sage: [(p,E.has_multiplicative_reduction(p)) for p in prime_range(15)]
            [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_multiplicative_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
        """
        return self.local_data(P).has_multiplicative_reduction()

    def has_split_multiplicative_reduction(self, P):
        r"""
        Return True if this elliptic curve has (bad) split multiplicative reduction at the prime `P`.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
          element generating such an ideal.

        OUTPUT:

        (bool) True if the curve has split multiplicative reduction at
        `P`, else False.

        EXAMPLES::

            sage: E=EllipticCurve('14a1')
            sage: [(p,E.has_split_multiplicative_reduction(p)) for p in prime_range(15)]
            [(2, False), (3, False), (5, False), (7, True), (11, False), (13, False)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_split_multiplicative_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
        """
        return self.local_data(P).has_split_multiplicative_reduction()

    def has_nonsplit_multiplicative_reduction(self, P):
        r"""
        Return True if this elliptic curve has (bad) non-split multiplicative reduction at the prime `P`.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
          element generating such an ideal.

        OUTPUT:

        (bool) True if the curve has non-split multiplicative
        reduction at `P`, else False.

        EXAMPLES::

            sage: E=EllipticCurve('14a1')
            sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in prime_range(15)]
            [(2, True), (3, False), (5, False), (7, False), (11, False), (13, False)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_nonsplit_multiplicative_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)]
        """
        return self.local_data(P).has_nonsplit_multiplicative_reduction()

    def has_additive_reduction(self, P):
        r"""
        Return True if this elliptic curve has (bad) additive reduction at the prime `P`.

        INPUT:

        - ``P`` -- a prime ideal of the base field of self, or a field
          element generating such an ideal.

        OUTPUT:

        (bool) True if the curve has additive reduction at `P`, else False.

        EXAMPLES::

            sage: E=EllipticCurve('27a1')
            sage: [(p,E.has_additive_reduction(p)) for p in prime_range(15)]
            [(2, False), (3, True), (5, False), (7, False), (11, False), (13, False)]

            sage: K.<a>=NumberField(x^3-2)
            sage: P17a, P17b = [P for P,e in K.factor(17)]
            sage: E = EllipticCurve([0,0,0,0,2*a+1])
            sage: [(p,E.has_additive_reduction(p)) for p in [P17a,P17b]]
            [(Fractional ideal (4*a^2 - 2*a + 1), False),
            (Fractional ideal (2*a + 1), True)]
        """
        return self.local_data(P).has_additive_reduction()

    def tamagawa_number(self, P, proof = None):
        r"""
        Returns the Tamagawa number of this elliptic curve at the prime `P`.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        OUTPUT:

        (positive integer) The Tamagawa number of the curve at `P`.

        EXAMPLES::

            sage: K.<a>=NumberField(x^2-5)
            sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
            sage: [E.tamagawa_number(P) for P in E.discriminant().support()]
            [1, 1, 1, 1]
            sage: K.<a> = QuadraticField(-11)
            sage: E = EllipticCurve('11a1').change_ring(K)
            sage: [E.tamagawa_number(P) for P in K(11).support()]
            [10]
        """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        return self.local_data(P, proof).tamagawa_number()

    def tamagawa_numbers(self):
        """
        Return a list of all Tamagawa numbers for all prime divisors of the
        conductor (in order).

        EXAMPLES::

            sage: e = EllipticCurve('30a1')
            sage: e.tamagawa_numbers()
            [2, 3, 1]
            sage: vector(e.tamagawa_numbers())
            (2, 3, 1)
            sage: K.<a>=NumberField(x^2+3)
            sage: eK = e.base_extend(K)
            sage: eK.tamagawa_numbers()
            [4, 6, 1]
        """
        return [self.tamagawa_number(p) for p in prime_divisors(self.conductor())]

    def tamagawa_exponent(self, P, proof = None):
        r"""
        Returns the Tamagawa index of this elliptic curve at the prime `P`.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        OUTPUT:

        (positive integer) The Tamagawa index of the curve at P.

        EXAMPLES::

            sage: K.<a>=NumberField(x^2-5)
            sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
            sage: [E.tamagawa_exponent(P) for P in E.discriminant().support()]
            [1, 1, 1, 1]
            sage: K.<a> = QuadraticField(-11)
            sage: E = EllipticCurve('11a1').change_ring(K)
            sage: [E.tamagawa_exponent(P) for P in K(11).support()]
            [10]
        """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        return self.local_data(P, proof).tamagawa_exponent()

    def tamagawa_product_bsd(self):
        r"""
        Given an elliptic curve `E` over a number field `K`, this function returns the
        integer `C(E/K)` that appears in the Birch and Swinnerton-Dyer conjecture accounting
        for the local information at finite places. If the model is a global minimal model then `C(E/K)` is
        simply the product of the Tamagawa numbers `c_v` where `v` runs over all prime ideals of `K`. Otherwise, if the model has to be changed at a place `v` a correction factor appears.
        The definition is such that `C(E/K)` times the periods at the infinite places is invariant
        under change of the Weierstrass model. See [Ta2] and [Do] for details.

        .. note::

            This definition is slightly different from the definition of ``tamagawa_product``
            for curves defined over `\QQ`. Over the rational number it is always defined to be the product
            of the Tamagawa numbers, so the two definitions only agree when the model is global minimal.

        OUTPUT:

        A rational number

        EXAMPLES::

            sage: K.<i> = NumberField(x^2+1)
            sage: E = EllipticCurve([0,2+i])
            sage: E.tamagawa_product_bsd()
            1

            sage: E = EllipticCurve([(2*i+1)^2,i*(2*i+1)^7])
            sage: E.tamagawa_product_bsd()
            4

        An example where the Neron model changes over K::

            sage: K.<t> = NumberField(x^5-10*x^3+5*x^2+10*x+1)
            sage: E = EllipticCurve(K,'75a1')
            sage: E.tamagawa_product_bsd()
            5
            sage: da = E.local_data()
            sage: [dav.tamagawa_number() for dav in da]
            [1, 1]

        An example over `\mathbb{Q}` (:trac:`9413`)::

            sage: E = EllipticCurve('30a')
            sage: E.tamagawa_product_bsd()
            6

        REFERENCES:

        - [Ta2] Tate, John, On the conjectures of Birch and Swinnerton-Dyer and a geometric analog. Seminaire Bourbaki, Vol. 9, Exp. No. 306.

        - [Do] Dokchitser, Tim and Vladimir, On the Birch-Swinnerton-Dyer quotients modulo squares, Annals of Math., 2010.

        """
        da = self.local_data()
        pr = 1
        for dav in da:
            pp = dav.prime()
            cv = dav.tamagawa_number()
            # uu is the quotient of the Neron differential at pp divided by
            # the differential associated to this particular equation E
            uu = self.isomorphism_to(dav.minimal_model()).u
            if self.base_field().absolute_degree() == 1:
                p = pp.gens_reduced()[0]
                f = 1
                v = valuation(ZZ(uu),p)
            else:
                p = pp.smallest_integer()
                f = pp.residue_class_degree()
                v = valuation(uu,pp)
            uu_abs_val = p**(f*v)
            pr *= cv * uu_abs_val
        return pr

    def kodaira_symbol(self, P, proof = None):
        r"""
        Returns the Kodaira Symbol of this elliptic curve at the prime `P`.

        INPUT:

        - ``P`` -- either None or a prime ideal of the base field of self.

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        OUTPUT:

        The Kodaira Symbol of the curve at P, represented as a string.

        EXAMPLES::

            sage: K.<a>=NumberField(x^2-5)
            sage: E=EllipticCurve([20, 225, 750, 625*a + 6875, 31250*a + 46875])
            sage: bad_primes = E.discriminant().support(); bad_primes
            [Fractional ideal (-a), Fractional ideal (7/2*a - 81/2), Fractional ideal (-a - 52), Fractional ideal (2)]
            sage: [E.kodaira_symbol(P) for P in bad_primes]
            [I0, I1, I1, II]
            sage: K.<a> = QuadraticField(-11)
            sage: E = EllipticCurve('11a1').change_ring(K)
            sage: [E.kodaira_symbol(P) for P in K(11).support()]
            [I10]
        """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        return self.local_data(P, proof).kodaira_symbol()


    def conductor(self):
        r"""
        Returns the conductor of this elliptic curve as a fractional
        ideal of the base field.

        OUTPUT:

        (fractional ideal) The conductor of the curve.

        EXAMPLES::

            sage: K.<i>=NumberField(x^2+1)
            sage: EllipticCurve([i, i - 1, i + 1, 24*i + 15, 14*i + 35]).conductor()
            Fractional ideal (21*i - 3)
            sage: K.<a>=NumberField(x^2-x+3)
            sage: EllipticCurve([1 + a , -1 + a , 1 + a , -11 + a , 5 -9*a  ]).conductor()
            Fractional ideal (-6*a)

        A not so well known curve with everywhere good reduction::

            sage: K.<a>=NumberField(x^2-38)
            sage: E=EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300])
            sage: E.conductor()
            Fractional ideal (1)

        An example which used to fail (see :trac:`5307`)::

            sage: K.<w>=NumberField(x^2+x+6)
            sage: E=EllipticCurve([w,-1,0,-w-6,0])
            sage: E.conductor()
            Fractional ideal (86304, w + 5898)

        An example raised in :trac:`11346`::

            sage: K.<g> = NumberField(x^2 - x - 1)
            sage: E1 = EllipticCurve(K,[0,0,0,-1/48,-161/864])
            sage: [(p.smallest_integer(),e) for p,e in E1.conductor().factor()]
            [(2, 4), (3, 1), (5, 1)]
        """
        try:
            return self._conductor
        except AttributeError:
            pass

        # Note: for number fields other than QQ we could initialize
        # N=K.ideal(1) or N=OK.ideal(1), which are the same, but for
        # K==QQ it has to be ZZ.ideal(1).
        OK = self.base_ring().ring_of_integers()
        self._conductor = prod([d.prime()**(d.conductor_valuation()) \
                                for d in self.local_data()],\
                               OK.ideal(1))
        return self._conductor

    def minimal_discriminant_ideal(self):
        r"""
        Return the minimal discriminant ideal of this elliptic curve.

        OUTPUT:

        The integral ideal `D` whose valuation at every prime `P` is
        that of the local minimal model for `E` at `P`.  If `E` has a
        global minimal model, this will be the principal ideal
        generated by the discriminant of any such model, bt otherwise
        it can be a proper divisor of the discrimanant of any model.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-x-57)
            sage: K.class_number()
            3
            sage: E = EllipticCurve([a,-a,a,-5692-820*a,-259213-36720*a])
            sage: K.ideal(E.discriminant())
            Fractional ideal (90118662980*a + 636812084644)
            sage: K.ideal(E.discriminant()).factor()
            (Fractional ideal (2))^2 * (Fractional ideal (3, a + 2))^12

        Here the minimal discriminant ideal is principal but there is
        no global minimal model since the quotient is the 12th power
        of a non-principal ideal::

            sage: E.minimal_discriminant_ideal()
            Fractional ideal (4)
            sage: E.minimal_discriminant_ideal().factor()
            (Fractional ideal (2))^2

        If (and only if) the curve has everywhere good reduction the
        result is the unit ideal::

            sage: K.<a> = NumberField(x^2-26)
            sage: E = EllipticCurve([a,a-1,a+1,4*a+10,2*a+6])
            sage: E.conductor()
            Fractional ideal (1)
            sage: E.discriminant()
            -104030*a - 530451
            sage: E.minimal_discriminant_ideal()
            Fractional ideal (1)

        Over `\QQ`, the result returned is an ideal of `\ZZ` rather
        than a fractional ideal of `\QQ`::

            sage: E = EllipticCurve([1,2,3,4,5])
            sage: E.minimal_discriminant_ideal()
            Principal ideal (10351) of Integer Ring
        """
        dat = self.local_data()
        # we treat separately the case where there are no bad primes,
        # which cannot happen over QQ, since ideals of QQ behave
        # differently to (fractional) ideals of other number fields.
        if len(dat)==0:
            return self.base_field().ideal(1)
        return prod([d.prime()**d.discriminant_valuation() for d in dat])

    def non_minimal_primes(self):
        r"""
        Returns a list of primes at which this elliptic curve is not minimal.

        OUTPUT:

        A list of prime ideals (or prime numbers when the base field
        is `\QQ`, empty if this is a global minimal model.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-10)
            sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a])
            sage: E.non_minimal_primes()
            [Fractional ideal (2, a), Fractional ideal (5, a)]
            sage: K.ideal(E.discriminant()).factor()
            (Fractional ideal (2, a))^24 * (Fractional ideal (3, a + 1))^5 * (Fractional ideal (3, a + 2))^5 * (Fractional ideal (5, a))^24 * (Fractional ideal (7))
            sage: E.minimal_discriminant_ideal().factor()
            (Fractional ideal (2, a))^12 * (Fractional ideal (3, a + 1))^5 * (Fractional ideal (3, a + 2))^5 * (Fractional ideal (7))

        Over `\QQ`, the primes returned are integers, not ideals::

            sage: E = EllipticCurve([0,0,0,-3024,46224])
            sage: E.non_minimal_primes()
            [2, 3]
            sage: Emin = E.global_minimal_model()
            sage: Emin.non_minimal_primes()
            []

        If the model is not globally integral, a ``ValueError`` is
        raised::

            sage: E = EllipticCurve([0,0,0,1/2,1/3])
            sage: E.non_minimal_primes()
            Traceback (most recent call last):
            ...
            ValueError: non_minimal_primes only defined for integral models
        """
        if not self.is_global_integral_model():
            raise ValueError("non_minimal_primes only defined for integral models")
        dat = self.local_data()
        D = self.discriminant()
        primes = [d.prime() for d in dat]
        if self.base_field() is QQ:
            primes = [P.gen() for P in primes]
        vals = [d.discriminant_valuation() for d in dat]
        return [P for P,v in zip(primes,vals) if D.valuation(P) > v]

    def is_global_minimal_model(self):
        r"""
        Returns whether this elliptic curve is a global minimal model.

        OUTPUT:

        Boolean, False if E is not integral, or if E is non-minimal at
        some prime, else True.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-10)
            sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a])
            sage: E.is_global_minimal_model()
            False
            sage: E.non_minimal_primes()
            [Fractional ideal (2, a), Fractional ideal (5, a)]

            sage: E = EllipticCurve([0,0,0,-3024,46224])
            sage: E.is_global_minimal_model()
            False
            sage: E.non_minimal_primes()
            [2, 3]
            sage: Emin = E.global_minimal_model()
            sage: Emin.is_global_minimal_model()
            True

        A necessary condition to be a global minimal model is that the
        model must be globally integral::

            sage: E = EllipticCurve([0,0,0,1/2,1/3])
            sage: E.is_global_minimal_model()
            False
            sage: Emin.is_global_minimal_model()
            True
            sage: Emin.ainvs()
            (0, 1, 1, -2, 0)
        """
        if not self.is_global_integral_model():
            return False
        return self.non_minimal_primes() == []

    def global_minimality_class(self):
        r"""
        Returns the obstruction to this curve having a global minimal model.

        OUTPUT:

        An ideal class of the base number field, which is trivial if
        and only if the elliptic curve has a global minimal model, and
        which can be used to find global and semi-global minimal
        models.

        EXAMPLES:

        A curve defined over a field of class number 2 with no global
        minimal model was a nontrivial minimality class::

            sage: K.<a> = NumberField(x^2-10)
            sage: K.class_number()
            2
            sage: E = EllipticCurve([0, 0, 0, -22500, 750000*a])
            sage: E.global_minimality_class()
            Fractional ideal class (10, 5*a)
            sage: E.global_minimality_class().order()
            2

        Over the same field, a curve defined by a non-minimal model
        has trivial class, showing that a global minimal model does
        exist::

            sage: K.<a> = NumberField(x^2-10)
            sage: E = EllipticCurve([0,0,0,4536*a+14148,-163728*a- 474336])
            sage: E.is_global_minimal_model()
            False
            sage: E.global_minimality_class()
            Trivial principal fractional ideal class

        Over a field of class number 1 the result is always the
        trivial class::

            sage: K.<a> = NumberField(x^2-5)
            sage: E = EllipticCurve([0, 0, 0, K(16), K(64)])
            sage: E.global_minimality_class()
            Trivial principal fractional ideal class

            sage: E = EllipticCurve([0, 0, 0, 16, 64])
            sage: E.base_field()
            Rational Field
            sage: E.global_minimality_class()
            1
        """
        K = self.base_field()
        Cl = K.class_group()
        if K.class_number()==1:
            return Cl(1)
        D = self.discriminant()
        dat = self.local_data()
        primes = [d.prime() for d in dat]
        vals = [d.discriminant_valuation() for d in dat]
        I = prod([P**((D.valuation(P)-v)//12) for P,v in zip(primes,vals)],
                 K.ideal(1))
        return Cl(I)

    def has_global_minimal_model(self):
        r"""
        Returns whether this elliptic curve has a global minimal model.

        OUTPUT:

        Boolean, True iff a global minimal model exists, i.e. an
        integral model which is minimal at every prime.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-10)
            sage: E = EllipticCurve([0,0,0,4536*a+14148,-163728*a-474336])
            sage: E.is_global_minimal_model()
            False
            sage: E.has_global_minimal_model()
            True
        """
        return self.global_minimality_class().is_one()

    def global_minimal_model(self, proof = None, semi_global=False):
        r"""
        Returns a model of self that is integral, and minimal.

        .. note::

           Over fields of class number greater than 1, a global
           minimal model may not exist.  If it does not, set the
           parameter ``semi_global`` to ``True`` to obtain a model
           minimal at all but one prime.

        INPUT:

        - ``proof`` -- whether to only use provably correct methods
          (default controlled by global proof module).  Note that the
          proof module is number_field, not elliptic_curves, since the
          functions that actually need the flag are in number fields.

        - ``semi_global`` (boolean, default False) -- if there is no
          global minimal mode, return a semi-global minimal model
          (minimal at all but one prime) instead, if True; raise an
          error if False.  No effect if a global minimal model exists.

        OUTPUT:

        A global integral and minimal model, or an integral model
        minimal at all but one prime of there is no global minimal
        model and the flag ``semi_global`` is True.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2-38)
            sage: E = EllipticCurve([0,0,0, 21796814856932765568243810*a - 134364590724198567128296995, 121774567239345229314269094644186997594*a - 750668847495706904791115375024037711300])
            sage: E2 = E.global_minimal_model()
            sage: E2
            Elliptic Curve defined by y^2 + a*x*y + (a+1)*y = x^3 + (a+1)*x^2 + (4*a+15)*x + (4*a+21) over Number Field in a with defining polynomial x^2 - 38
            sage: E2.local_data()
            []

        See :trac:`11347`::

            sage: K.<g> = NumberField(x^2 - x - 1)
            sage: E = EllipticCurve(K,[0,0,0,-1/48,161/864]).integral_model().global_minimal_model(); E
            Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 over Number Field in g with defining polynomial x^2 - x - 1
            sage: [(p.norm(), e) for p, e in E.conductor().factor()]
            [(9, 1), (5, 1)]
            sage: [(p.norm(), e) for p, e in E.discriminant().factor()]
            [(-5, 2), (9, 1)]

        See :trac:`14472`, this used not to work over a relative extension::

            sage: K1.<w> = NumberField(x^2+x+1)
            sage: m = polygen(K1)
            sage: K2.<v> = K1.extension(m^2-w+1)
            sage: E = EllipticCurve([0*v,-432])
            sage: E.global_minimal_model()
            Elliptic Curve defined by y^2 + (v+w+1)*y = x^3 + ((6*w-10)*v+16*w+20) over Number Field in v with defining polynomial x^2 - w + 1 over its base field

        See :trac:`18662`: for fields of class number greater than 1,
        even when global minimal models did exist, their computation
        was not implemented.  Now it is::

            sage: K.<a> = NumberField(x^2-10)
            sage: K.class_number()
            2
            sage: E = EllipticCurve([0,0,0,-186408*a - 589491, 78055704*a + 246833838])
            sage: E.discriminant().norm()
            16375845905239507992576
            sage: E.discriminant().norm().factor()
            2^31 * 3^27
            sage: E.has_global_minimal_model()
            True
            sage: Emin = E.global_minimal_model(); Emin
            Elliptic Curve defined by y^2 + (a+1)*x*y + (a+1)*y = x^3 + (-a)*x^2 + (a-12)*x + (-2*a+2) over Number Field in a with defining polynomial x^2 - 10
            sage: Emin.discriminant().norm()
            3456
            sage: Emin.discriminant().norm().factor()
            2^7 * 3^3

        If there is no global minimal model, this method will raise an
        error unless you set the parameter ``semi_global`` to ``True``::

            sage: K.<a> = NumberField(x^2-10)
            sage: K.class_number()
            2
            sage: E = EllipticCurve([a,a,0,3*a+8,4*a+3])
            sage: E.has_global_minimal_model()
            False
            sage: E.global_minimal_model()
            Traceback (most recent call last):
            ...
            ValueError: Elliptic Curve defined by y^2 + a*x*y = x^3 + a*x^2 + (3*a+8)*x + (4*a+3) over Number Field in a with defining polynomial x^2 - 10 has no global minimal model!  For a semi-global minimal model use semi_global=True
            sage: E.global_minimal_model(semi_global=True)
            Elliptic Curve defined by y^2 + a*x*y = x^3 + a*x^2 + (3*a+8)*x + (4*a+3) over Number Field in a with defining polynomial x^2 - 10

        An example of a curve with everywhere good reduction but which
        has no model with unit discriminant::

            sage: K.<a> = NumberField(x^2-x-16)
            sage: K.class_number()
            2
            sage: E = EllipticCurve([0,0,0,-15221331*a - 53748576, -79617688290*a - 281140318368])
            sage: Emin = E.global_minimal_model(semi_global=True)
            sage: Emin.ainvs()
            (a, a - 1, a, 605*a - 2728, 15887*a - 71972)
            sage: Emin.discriminant()
            -17*a - 16
            sage: Emin.discriminant().norm()
            -4096
            sage: Emin.minimal_discriminant_ideal()
            Fractional ideal (1)
            sage: E.conductor()
            Fractional ideal (1)
       """
        if proof is None:
            import sage.structure.proof.proof
            # We use the "number_field" flag because the actual proof dependence is in PARI's number field functions.
            proof = sage.structure.proof.proof.get_flag(None, "number_field")

        if self.has_global_minimal_model() or semi_global:
            if self.base_ring().class_number() == 1:
                E = self.global_integral_model()
                for P in E.base_ring()(E.discriminant()).support():
                    E = E.local_data(P,proof, globally=True).minimal_model()
            else:
                from kraus import semi_global_minimal_model
                E, P = semi_global_minimal_model(self)
            return E._scale_by_units()._reduce_model()

        raise ValueError("%s has no global minimal model!  For a semi-global minimal model use semi_global=True" % self)

    def reduction(self,place):
       r"""
       Return the reduction of the elliptic curve at a place of good reduction.

       INPUT:

       - ``place`` -- a prime ideal in the base field of the curve

       OUTPUT:

       An elliptic curve over a finite field, the residue field of the place.

       EXAMPLES::

           sage: K.<i> = QuadraticField(-1)
           sage: EK = EllipticCurve([0,0,0,i,i+3])
           sage: v = K.fractional_ideal(2*i+3)
           sage: EK.reduction(v)
           Elliptic Curve defined by y^2  = x^3 + 5*x + 8 over Residue field of Fractional ideal (2*i + 3)
           sage: EK.reduction(K.ideal(1+i))
           Traceback (most recent call last):
           ...
           ValueError: The curve must have good reduction at the place.
           sage: EK.reduction(K.ideal(2))
           Traceback (most recent call last):
           ...
           ValueError: The ideal must be prime.
           sage: K=QQ.extension(x^2+x+1,"a")
           sage: E=EllipticCurve([1024*K.0,1024*K.0])
           sage: E.reduction(2*K)
           Elliptic Curve defined by y^2 + (abar+1)*y = x^3 over Residue field in abar of Fractional ideal (2)
       """
       K = self.base_field()
       OK = K.ring_of_integers()
       try:
           place = K.ideal(place)
       except TypeError:
           raise TypeError("The parameter must be an ideal of the base field of the elliptic curve")
       if not place.is_prime():
           raise ValueError("The ideal must be prime.")
       disc = self.discriminant()
       if not K.ideal(disc).valuation(place) == 0:
           local_data=self.local_data(place)
           if local_data.has_good_reduction():
               Fv = OK.residue_field(place)
               return local_data.minimal_model().change_ring(Fv)
           raise ValueError("The curve must have good reduction at the place.")
       Fv = OK.residue_field(place)
       return self.change_ring(Fv)

    def _torsion_bound(self,number_of_places = 20):
        r"""
        An upper bound on the order of the torsion subgroup.

        INPUT:

        - ``number_of_places`` (positive integer, default = 20) -- the
          number of places that will be used to find the bound.

        OUTPUT:

        (integer) An upper bound on the torsion order.

        ALGORITHM:

        An upper bound on the order of the torsion.group of the
        elliptic curve is obtained by counting points modulo several
        primes of good reduction.  Note that the upper bound returned
        by this function is a multiple of the order of the torsion
        group, and in general will be greater than the order.

        EXAMPLES::

            sage: CDB=CremonaDatabase()
            sage: [E._torsion_bound() for E in CDB.iter([14])]
            [6, 6, 6, 6, 6, 6]
            sage: [E.torsion_order() for E in CDB.iter([14])]
            [6, 6, 2, 6, 2, 6]

        An example over a relative number field (see :trac:`16011`)::

            sage: R.<x> = QQ[]
            sage: F.<a> = QuadraticField(5)
            sage: K.<b> = F.extension(x^2-3)
            sage: E = EllipticCurve(K,[0,0,0,b,1])
            sage: E.torsion_subgroup().order()
            1

        """
        E = self
        bound = 0
        k = 0
        K = E.base_field()
        OK = K.ring_of_integers()
        disc = E.discriminant()
        p = Integer(1)
        # runs through primes, decomposes them into prime ideals
        while k < number_of_places :
            p = p.next_prime()
            f = K.primes_above(p)
            # runs through prime ideals above p
            for qq in f:
                fqq = qq.residue_class_degree()
                charqq = qq.smallest_integer()
                # take only places with small residue field (so that the
                # number of points will be small)
                if fqq == 1 or charqq**fqq < 3*number_of_places:
                    # check if the model is integral at the place
                    if min([K.ideal(a).valuation(qq) for a in E.a_invariants()]) >= 0:
                        eqq = qq.absolute_ramification_index()
                        # check if the formal group at the place is torsion-free
                        # if so the torsion injects into the reduction
                        if eqq < charqq - 1 and disc.valuation(qq) == 0:
                            Etilda = E.reduction(qq)
                            Npp = Etilda.cardinality()
                            bound = gcd(bound,Npp)
                            if bound == 1:
                                return bound
                            k += 1
        return bound

    def torsion_subgroup(self):
        r"""
        Returns the torsion subgroup of this elliptic curve.

        OUTPUT:

        (``EllipticCurveTorsionSubgroup``) The
        ``EllipticCurveTorsionSubgroup`` associated to this elliptic
        curve.

        EXAMPLES::

            sage: E = EllipticCurve('11a1')
            sage: K.<t>=NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
            sage: EK = E.base_extend(K)
            sage: tor = EK.torsion_subgroup()  # long time (2s on sage.math, 2014)
            sage: tor  # long time
            Torsion Subgroup isomorphic to Z/5 + Z/5 associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in t with defining polynomial x^4 + x^3 + 11*x^2 + 41*x + 101
            sage: tor.gens()  # long time
            ((16 : 60 : 1), (t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1))

        ::

            sage: E = EllipticCurve('15a1')
            sage: K.<t>=NumberField(x^2 + 2*x + 10)
            sage: EK=E.base_extend(K)
            sage: EK.torsion_subgroup()
            Torsion Subgroup isomorphic to Z/4 + Z/4 associated to the Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 + (-10)*x + (-10) over Number Field in t with defining polynomial x^2 + 2*x + 10

        ::

            sage: E = EllipticCurve('19a1')
            sage: K.<t>=NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1)
            sage: EK=E.base_extend(K)
            sage: EK.torsion_subgroup()
            Torsion Subgroup isomorphic to Z/9 associated to the Elliptic Curve defined by y^2 + y = x^3 + x^2 + (-9)*x + (-15) over Number Field in t with defining polynomial x^9 - 3*x^8 - 4*x^7 + 16*x^6 - 3*x^5 - 21*x^4 + 5*x^3 + 7*x^2 - 7*x + 1

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: EK = EllipticCurve([0,0,0,i,i+3])
            sage: EK.torsion_subgroup ()
            Torsion Subgroup isomorphic to Trivial group associated to the Elliptic Curve defined by y^2  = x^3 + i*x + (i+3) over Number Field in i with defining polynomial x^2 + 1
        """
        try:
            return self.__torsion_subgroup
        except AttributeError:
            self.__torsion_subgroup = ell_torsion.EllipticCurveTorsionSubgroup(self)
            return self.__torsion_subgroup

    def torsion_order(self):
        r"""
        Returns the order of the torsion subgroup of this elliptic curve.

        OUTPUT:

        (integer) the order of the torsion subgroup of this elliptic curve.

        EXAMPLES::

            sage: E = EllipticCurve('11a1')
            sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
            sage: EK = E.base_extend(K)
            sage: EK.torsion_order()  # long time (2s on sage.math, 2014)
            25

        ::

            sage: E = EllipticCurve('15a1')
            sage: K.<t> = NumberField(x^2 + 2*x + 10)
            sage: EK = E.base_extend(K)
            sage: EK.torsion_order()
            16

        ::

            sage: E = EllipticCurve('19a1')
            sage: K.<t> = NumberField(x^9-3*x^8-4*x^7+16*x^6-3*x^5-21*x^4+5*x^3+7*x^2-7*x+1)
            sage: EK = E.base_extend(K)
            sage: EK.torsion_order()
            9

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: EK = EllipticCurve([0,0,0,i,i+3])
            sage: EK.torsion_order()
            1
         """
        try:
            return self.__torsion_order
        except AttributeError:
            self.__torsion_order = self.torsion_subgroup().order()
            return self.__torsion_order

    def torsion_points(self):
        r"""
        Returns a list of the torsion points of this elliptic curve.

        OUTPUT:

        (list) A sorted list of the torsion points.

        EXAMPLES::

            sage: E = EllipticCurve('11a1')
            sage: E.torsion_points()
            [(0 : 1 : 0), (5 : -6 : 1), (5 : 5 : 1), (16 : -61 : 1), (16 : 60 : 1)]
            sage: K.<t> = NumberField(x^4 + x^3 + 11*x^2 + 41*x + 101)
            sage: EK = E.base_extend(K)
            sage: EK.torsion_points()  # long time (1s on sage.math, 2014)
            [(16 : 60 : 1),
             (5 : 5 : 1),
             (5 : -6 : 1),
             (16 : -61 : 1),
             (t : 1/11*t^3 + 6/11*t^2 + 19/11*t + 48/11 : 1),
             (-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : 6/55*t^3 + 3/55*t^2 + 25/11*t + 156/55 : 1),
             (-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : -7/121*t^3 + 24/121*t^2 + 197/121*t + 16/121 : 1),
             (5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : -49/121*t^3 - 129/121*t^2 - 315/121*t - 207/121 : 1),
             (10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : 32/121*t^3 + 60/121*t^2 - 261/121*t - 807/121 : 1),
             (1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : -6/11*t^3 - 3/11*t^2 - 26/11*t - 321/11 : 1),
             (14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : 16/121*t^3 - 69/121*t^2 + 293/121*t - 46/121 : 1),
             (3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : 7/55*t^3 - 24/55*t^2 + 9/11*t + 17/55 : 1),
             (-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : 34/121*t^3 - 27/121*t^2 + 305/121*t + 708/121 : 1),
             (-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : 15/121*t^3 + 156/121*t^2 - 232/121*t + 2766/121 : 1),
             (1/11*t^3 - 5/11*t^2 + 19/11*t - 40/11 : 6/11*t^3 + 3/11*t^2 + 26/11*t + 310/11 : 1),
             (-26/121*t^3 + 20/121*t^2 - 219/121*t - 995/121 : -15/121*t^3 - 156/121*t^2 + 232/121*t - 2887/121 : 1),
             (-5/121*t^3 + 36/121*t^2 - 84/121*t + 24/121 : -34/121*t^3 + 27/121*t^2 - 305/121*t - 829/121 : 1),
             (3/55*t^3 + 7/55*t^2 + 2/55*t + 78/55 : -7/55*t^3 + 24/55*t^2 - 9/11*t - 72/55 : 1),
             (14/121*t^3 - 15/121*t^2 + 90/121*t + 232/121 : -16/121*t^3 + 69/121*t^2 - 293/121*t - 75/121 : 1),
             (t : -1/11*t^3 - 6/11*t^2 - 19/11*t - 59/11 : 1),
             (10/121*t^3 + 49/121*t^2 + 168/121*t + 73/121 : -32/121*t^3 - 60/121*t^2 + 261/121*t + 686/121 : 1),
             (5/121*t^3 - 14/121*t^2 - 158/121*t - 453/121 : 49/121*t^3 + 129/121*t^2 + 315/121*t + 86/121 : 1),
             (-9/121*t^3 - 21/121*t^2 - 127/121*t - 377/121 : 7/121*t^3 - 24/121*t^2 - 197/121*t - 137/121 : 1),
             (-3/55*t^3 - 7/55*t^2 - 2/55*t - 133/55 : -6/55*t^3 - 3/55*t^2 - 25/11*t - 211/55 : 1),
             (0 : 1 : 0)]

        ::

            sage: E = EllipticCurve('15a1')
            sage: K.<t> = NumberField(x^2 + 2*x + 10)
            sage: EK = E.base_extend(K)
            sage: EK.torsion_points()
            [(-7 : -5*t - 2 : 1),
             (-7 : 5*t + 8 : 1),
             (-13/4 : 9/8 : 1),
             (-2 : -2 : 1),
             (-2 : 3 : 1),
             (-t - 2 : -t - 7 : 1),
             (-t - 2 : 2*t + 8 : 1),
             (-1 : 0 : 1),
             (t : t - 5 : 1),
             (t : -2*t + 4 : 1),
             (0 : 1 : 0),
             (1/2 : -5/4*t - 2 : 1),
             (1/2 : 5/4*t + 1/2 : 1),
             (3 : -2 : 1),
             (8 : -27 : 1),
             (8 : 18 : 1)]

        ::

            sage: K.<i> = QuadraticField(-1)
            sage: EK = EllipticCurve(K,[0,0,0,0,-1])
            sage: EK.torsion_points ()
             [(-2 : -3*i : 1), (-2 : 3*i : 1), (0 : -i : 1), (0 : i : 1), (0 : 1 : 0), (1 : 0 : 1)]
         """
        T = self.torsion_subgroup() # make sure it is cached
        return sorted(T.points())           # these are also cached in T

    def rank_bounds(self, **kwds):
        r"""
        Returns the lower and upper bounds using :meth:`~simon_two_descent`.
        The results of :meth:`~simon_two_descent` are cached.

        .. NOTE::

            The optional parameters control the Simon two descent algorithm;
            see the documentation of :meth:`~simon_two_descent` for more
            details.

        INPUT:

        - ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level

        - ``lim1``    -- (default: 2) limit on trivial points on quartics

        - ``lim3``  -- (default: 4) limit on points on ELS quartics

        - ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve

        - ``maxprob`` -- (default: 20)

        - ``limbigprime`` -- (default: 30) to distinguish between
          small and large prime numbers. Use probabilistic tests for
          large primes. If 0, don't use probabilistic tests.

        - ``known_points`` -- (default: None) list of known points on
          the curve

        OUTPUT:

        lower and upper bounds for the rank of the Mordell-Weil group


        .. NOTE::

           For non-quadratic number fields, this code does
           return, but it takes a long time.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 23, 'a')
            sage: E = EllipticCurve(K, '37')
            sage: E == loads(dumps(E))
            True
            sage: E.rank_bounds()
            (2, 2)

        Here is a curve with two-torsion, again the bounds coincide::

            sage: Qrt5.<rt5>=NumberField(x^2-5)
            sage: E=EllipticCurve([0,5-rt5,0,rt5,0])
            sage: E.rank_bounds()
            (1, 1)

        Finally an example with non-trivial 2-torsion in Sha. So the
        2-descent will not be able to determine the rank, but can only
        give bounds::

            sage: E = EllipticCurve("15a5")
            sage: K.<t> = NumberField(x^2-6)
            sage: EK = E.base_extend(K)
            sage: EK.rank_bounds(lim1=1,lim3=1,limtriv=1)
            (0, 2)

        IMPLEMENTATION:

        Uses Denis Simon's PARI/GP scripts from
        http://www.math.unicaen.fr/~simon/.

        """
        lower, upper, gens = self.simon_two_descent(**kwds)
        # this was corrected in trac 13593. upper is the dimension
        # of the 2-selmer group, so we can certainly remove the
        # 2-torsion of the Mordell-Weil group.
        upper -= self.two_torsion_rank()
        return lower, upper

    def rank(self, **kwds):
        r"""
        Return the rank of this elliptic curve, if it can be determined.

        .. NOTE::

            The optional parameters control the Simon two descent algorithm;
            see the documentation of :meth:`~simon_two_descent` for more
            details.

        INPUT:

        - ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level

        - ``lim1``    -- (default: 2) limit on trivial points on quartics

        - ``lim3``  -- (default: 4) limit on points on ELS quartics

        - ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve

        - ``maxprob`` -- (default: 20)

        - ``limbigprime`` -- (default: 30) to distinguish between
          small and large prime numbers. Use probabilistic tests for
          large primes. If 0, don't use probabilistic tests.

        - ``known_points`` -- (default: None) list of known points on
          the curve

        OUTPUT:

        If the upper and lower bounds given by Simon two-descent are
        the same, then the rank has been uniquely identified and we
        return this. Otherwise, we raise a ValueError with an error
        message specifying the upper and lower bounds.

        .. NOTE::

           For non-quadratic number fields, this code does return, but it takes
           a long time.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 23, 'a')
            sage: E = EllipticCurve(K, '37')
            sage: E == loads(dumps(E))
            True
            sage: E.rank()
            2

        Here is a curve with two-torsion in the Tate-Shafarevich group,
        so here the bounds given by the algorithm do not uniquely
        determine the rank::

            sage: E = EllipticCurve("15a5")
            sage: K.<t> = NumberField(x^2-6)
            sage: EK = E.base_extend(K)
            sage: EK.rank(lim1=1, lim3=1, limtriv=1)
            Traceback (most recent call last):
            ...
            ValueError: There is insufficient data to determine the rank -
            2-descent gave lower bound 0 and upper bound 2

        IMPLEMENTATION:

        Uses Denis Simon's PARI/GP scripts from
        http://www.math.unicaen.fr/~simon/.

        """
        lower, upper = self.rank_bounds(**kwds)
        if lower == upper:
            return lower
        else:
            raise ValueError('There is insufficient data to determine the rank - 2-descent gave lower bound %s and upper bound %s' % (lower, upper))

    def gens(self, **kwds):
        r"""
        Return some points of infinite order on this elliptic curve.

        Contrary to what the name of this method suggests, the points
        it returns do not always generate a subgroup of full rank in
        the Mordell-Weil group, nor are they necessarily linearly
        independent.  Moreover, the number of points can be smaller or
        larger than what one could expect after calling :meth:`~rank`
        or :meth:`~rank_bounds`.

        .. NOTE::

            The optional parameters control the Simon two descent algorithm;
            see the documentation of :meth:`~simon_two_descent` for more
            details.

        INPUT:

        - ``verbose`` -- 0, 1, 2, or 3 (default: 0), the verbosity level

        - ``lim1``    -- (default: 2) limit on trivial points on quartics

        - ``lim3``  -- (default: 4) limit on points on ELS quartics

        - ``limtriv`` -- (default: 2) limit on trivial points on elliptic curve

        - ``maxprob`` -- (default: 20)

        - ``limbigprime`` -- (default: 30) to distinguish between
          small and large prime numbers. Use probabilistic tests for
          large primes. If 0, don't use probabilistic tests.

        - ``known_points`` -- (default: None) list of known points on
          the curve

        OUTPUT:

        A set of points of infinite order given by the Simon two-descent.

        .. NOTE::

           For non-quadratic number fields, this code does return, but it takes
           a long time.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 23, 'a')
            sage: E = EllipticCurve(K, '37')
            sage: E == loads(dumps(E))
            True
            sage: E.gens()
            [(0 : 0 : 1), (1/8*a + 5/8 : -3/16*a - 7/16 : 1)]

        It can happen that no points are found if the height bounds
        used in the search are too small (see :trac:`10745`)::

            sage: K.<y> = NumberField(x^4 + x^2 - 7)
            sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0])
            sage: E.gens(lim1=1, lim3=1)
            []
            sage: E.rank(), E.gens()  # long time (about 3 s)
            (1, [(9/25*y^2 + 26/25 : -229/125*y^3 - 67/25*y^2 - 731/125*y - 213/25 : 1)])

        Here is a curve of rank 2, yet the list contains many points::

            sage: K.<t> = NumberField(x^2-17)
            sage: E = EllipticCurve(K,[-4,0])
            sage: E.gens()
            [(-1/2*t + 1/2 : -1/2*t + 1/2 : 1),
            (-2*t + 8 : -8*t + 32 : 1),
            (1/2*t + 3/2 : -1/2*t - 7/2 : 1),
            (-1/8*t - 7/8 : -1/16*t - 23/16 : 1),
            (1/8*t - 7/8 : -1/16*t + 23/16 : 1),
            (t + 3 : -2*t - 10 : 1),
            (2*t + 8 : -8*t - 32 : 1),
            (1/2*t + 1/2 : -1/2*t - 1/2 : 1),
            (-1/2*t + 3/2 : -1/2*t + 7/2 : 1),
            (t + 7 : -4*t - 20 : 1),
            (-t + 7 : -4*t + 20 : 1),
            (-t + 3 : -2*t + 10 : 1)]
            sage: E.rank()
            2

        Test that points of finite order are not included (see :trac:`13593`)::

            sage: E = EllipticCurve("17a3")
            sage: K.<t> = NumberField(x^2+3)
            sage: EK = E.base_extend(K)
            sage: EK.rank()
            0
            sage: EK.gens()
            []

        IMPLEMENTATION:

        Uses Denis Simon's PARI/GP scripts from
        http://www.math.unicaen.fr/~simon/.
        """
        _ = self.simon_two_descent(**kwds)
        return self._known_points

    def period_lattice(self, embedding):
        r"""
        Returns the period lattice of the elliptic curve for the given
        embedding of its base field with respect to the differential
        `dx/(2y + a_1x + a_3)`.

        INPUT:

        - ``embedding`` - an embedding of the base number field into `\RR` or `\CC`.

        .. note::

           The precision of the embedding is ignored: we only use the
           given embedding to determine which embedding into ``QQbar``
           to use.  Once the lattice has been initialized, periods can
           be computed to arbitrary precision.


        EXAMPLES:

        First define a field with two real embeddings::

            sage: K.<a> = NumberField(x^3-2)
            sage: E=EllipticCurve([0,0,0,a,2])
            sage: embs=K.embeddings(CC); len(embs)
            3

        For each embedding we have a different period lattice::

            sage: E.period_lattice(embs[0])
            Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
            From: Number Field in a with defining polynomial x^3 - 2
            To:   Algebraic Field
            Defn: a |--> -0.6299605249474365? - 1.091123635971722?*I

            sage: E.period_lattice(embs[1])
            Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
            From: Number Field in a with defining polynomial x^3 - 2
            To:   Algebraic Field
            Defn: a |--> -0.6299605249474365? + 1.091123635971722?*I

            sage: E.period_lattice(embs[2])
            Period lattice associated to Elliptic Curve defined by y^2 = x^3 + a*x + 2 over Number Field in a with defining polynomial x^3 - 2 with respect to the embedding Ring morphism:
            From: Number Field in a with defining polynomial x^3 - 2
            To:   Algebraic Field
            Defn: a |--> 1.259921049894873?

        Although the original embeddings have only the default
        precision, we can obtain the basis with higher precision
        later::

            sage: L=E.period_lattice(embs[0])
            sage: L.basis()
            (1.86405007647981 - 0.903761485143226*I, -0.149344633143919 - 2.06619546272945*I)

            sage: L.basis(prec=100)
            (1.8640500764798108425920506200 - 0.90376148514322594749786960975*I, -0.14934463314391922099120107422 - 2.0661954627294548995621225062*I)
        """
        from sage.schemes.elliptic_curves.period_lattice import PeriodLattice_ell
        return PeriodLattice_ell(self,embedding)

    def height_function(self):
        """
        Return the canonical height function attached to self.

        EXAMPLE::

            sage: K.<a> = NumberField(x^2 - 5)
            sage: E = EllipticCurve(K, '11a3')
            sage: E.height_function()
            EllipticCurveCanonicalHeight object associated to Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 over Number Field in a with defining polynomial x^2 - 5

        """
        if not hasattr(self, '_height_function'):
            from sage.schemes.elliptic_curves.height import EllipticCurveCanonicalHeight
            self._height_function = EllipticCurveCanonicalHeight(self)
        return self._height_function

    ##########################################################
    # Isogeny class
    ##########################################################
    def isogeny_class(self):
        r"""
        Returns the isogeny class of this elliptic curve.

        OUTPUT:

        An instance of the class
        :class:`sage.schemes.elliptic_curves.isogeny_class.IsogenyClass_EC_NumberField`.
        From this object may be obtained a list of curves in the
        class, a matrix of the degrees of the isogenies between them,
        and the isogenies themselves.

        .. note::

            The curves in the isogeny class will all be minimal models
            if these exist (for example, when the class number is
            `1`); otherwise they will be minimal at all but one prime.

        EXAMPLES::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve(K, [0,0,0,0,1])
            sage: C = E.isogeny_class(); C
            Isogeny class of Elliptic Curve defined by y^2 = x^3 + 1 over Number Field in i with defining polynomial x^2 + 1

        The curves in the class (sorted)::

            sage: [E1.ainvs() for E1 in C]
            [(0, 0, 0, 0, -27),
            (0, 0, 0, 0, 1),
            (i + 1, i, i + 1, -i + 3, 4*i),
            (i + 1, i, i + 1, -i + 33, -58*i)]

        The matrix of degrees of cyclic isogenies between curves::

            sage: C.matrix()
            [1 3 6 2]
            [3 1 2 6]
            [6 2 1 3]
            [2 6 3 1]

        The array of isogenies themselves is not filled out but only
        contains those used to construct the class, the other entries
        containing the interger 0.  This will be changed when the
        class :class:`EllipticCurveIsogeny` allowed composition.  In
        this case we used `2`-isogenies to go from 0 to 2 and from 1
        to 3, and `3`-isogenies to go from 0 to 1 and from 2 to 3::

            sage: isogs = C.isogenies()
            sage: [((i,j),isogs[i][j].degree()) for i in range(4) for j in range(4) if isogs[i][j]!=0]
            [((0, 1), 3), ((2, 1), 2), ((3, 0), 2), ((3, 2), 3)]
            sage: [((i,j),isogs[i][j].x_rational_map()) for i in range(4) for j in range(4) if isogs[i][j]!=0]
            [((0, 1), (1/9*x^3 - 12)/x^2),
            ((2, 1), (-1/2*i*x^2 + x)/(x + 3/2*i)),
            ((3, 0), (-1/2*i*x^2 - x - 4*i)/(x - 5/2*i)),
            ((3, 2), (1/9*x^3 - 4/3*i*x^2 - 34/3*x + 226/9*i)/(x^2 - 8*i*x - 16))]

        The isogeny class may be visualized by obtaining its graph and
        plotting it::

            sage: G = C.graph()
            sage: G.show(edge_labels=True) # long time

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([1+i, -i, i, 1, 0])
            sage: C = E.isogeny_class(); C
            Isogeny class of Elliptic Curve defined by y^2 + (i+1)*x*y + i*y = x^3 + (-i)*x^2 + x over Number Field in i with defining polynomial x^2 + 1
            sage: len(C)
            6
            sage: C.matrix()
            [ 1  3  9 18  6  2]
            [ 3  1  3  6  2  6]
            [ 9  3  1  2  6 18]
            [18  6  2  1  3  9]
            [ 6  2  6  3  1  3]
            [ 2  6 18  9  3  1]
            sage: [E1.ainvs() for E1 in C]
            [(i + 1, i - 1, i, -i - 1, -i + 1),
            (i + 1, i - 1, i, 14*i + 4, 7*i + 14),
            (i + 1, i - 1, i, 59*i + 99, 372*i - 410),
            (i + 1, -i, i, -240*i - 399, 2869*i + 2627),
            (i + 1, -i, i, -5*i - 4, 2*i + 5),
            (i + 1, -i, i, 1, 0)]

        An example with CM by `\sqrt{-5}`::

            sage: pol = PolynomialRing(QQ,'x')([1,0,3,0,1])
            sage: K.<c> = NumberField(pol)
            sage: j = 1480640+565760*c^2
            sage: E = EllipticCurve(j=j)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            True
            sage: E.cm_discriminant()
            -20
            sage: C = E.isogeny_class()
            sage: len(C)
            2
            sage: C.matrix()
            [1 2]
            [2 1]
            sage: [E.ainvs() for E in C]
            [(0, 0, 0, -25762110*c^2 - 67447215, -154360009760*c^2 - 404119737340),
            (0, 0, 0, 130763490*c^2 + 342343485, 1391590873420*c^3 + 3643232206680*c)]
            sage: C.isogenies()[0][1]
            Isogeny of degree 2 from Elliptic Curve defined by y^2 = x^3 + (-25762110*c^2-67447215)*x + (-154360009760*c^2-404119737340) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1 to Elliptic Curve defined by y^2 = x^3 + (130763490*c^2+342343485)*x + (-1391590873420*c^3-3643232206680*c) over Number Field in c with defining polynomial x^4 + 3*x^2 + 1

        An example with CM by `\sqrt{-23}` (class number `3`)::

            sage: pol = PolynomialRing(QQ,'x')([1,-3,5,-5,5,-3,1])
            sage: L.<a> = NumberField(pol)
            sage: js = hilbert_class_polynomial(-23).roots(L,multiplicities=False); len(js)
            3
            sage: E = EllipticCurve(j=js[0])
            sage: E.has_rational_cm()
            True
            sage: len(E.isogenies_prime_degree())
            3
            sage: C = E.isogeny_class(); len(C)
            6

        The reason for the isogeny class having size six while the
        class number is only `3` is that the class also contains three
        curves with CM by the order of discriminant `-92=4\cdot(-23)`,
        which also has class number `3`.  The curves in the class are
        sorted first by CM discriminant (then lexicographically using
        a-invariants)::

            sage: [F.cm_discriminant() for F in C]
            [-23, -23, -23, -92, -92, -92]

        `2` splits in the order with discriminant `-23`, into two
        primes of order `3` in the class group, each of which induces
        a `2`-isogeny to a curve with the same endomorphism ring; the
        third `2`-isogeny is to a curve with the smaller endomorphism
        ring::

            sage: [phi.codomain().cm_discriminant() for phi in E.isogenies_prime_degree()]
            [-92, -23, -23]

            sage: C.matrix()
            [1 2 2 2 4 4]
            [2 1 2 4 2 4]
            [2 2 1 4 4 2]
            [2 4 4 1 3 3]
            [4 2 4 3 1 3]
            [4 4 2 3 3 1]

        The graph of this isogeny class has a shape which does not
        occur over `\QQ`: a triangular prism.  Note that for curves
        without CM, the graph has an edge between two curves if and
        only if they are connected by an isogeny of prime degree, and
        this degree is uniquely determined by the two curves, but in
        the CM case this property does not hold, since for pairs of
        curves in the class with the same endomorphism ring `O`, the
        set of degrees of isogenies between them is the set of
        integers represented by a primitive integral binary quadratic
        form of discriminant `\text{disc}(O)`, and this form
        represents infinitely many primes.  In the matrix we give a
        small prime represented by the appropriate form.  In this
        example, the matrix is formed by four `3\times3` blocks.  The
        isogenies of degree `2` indicated by the upper left `3\times3`
        block of the matrix could be replaced by isogenies of any
        degree represented by the quadratic form `2x^2+xy+3y^2` of
        discriminant `-23`.  Similarly in the lower right block, the
        entries of `3` could be represented by any integers
        represented by the quadratic form `3x^2+2xy+8y^2` of
        discriminant `-92`.  In the top right block and lower left
        blocks, by contrast, the prime entries `2` are unique
        determined::

            sage: G = C.graph()
            sage: G.adjacency_matrix()
            [0 1 1 1 0 0]
            [1 0 1 0 1 0]
            [1 1 0 0 0 1]
            [1 0 0 0 1 1]
            [0 1 0 1 0 1]
            [0 0 1 1 1 0]

        To display the graph without any edge labels::

            G.show() # long time

        To display the graph with edge labels: by default, for curves
        with rational CM, the labels are the coefficients of the
        associated quadratic forms::

            G.show(edge_labels=True) # long time

        For an alternative view, first relabel the edges using only 2
        labels to distinguish between isogenies between curves with
        the same endomorphism ring and isogenies between curves with
        different endomorphism rings, then use a 3-dimensional plot
        which can be rotated::

            sage: for i,j,l in G.edge_iterator():  G.set_edge_label(i,j,l.count(','))
            sage: G.show3d(color_by_label=True)

        A class number `6` example.  First we set up the fields: ``pol``
        defines the same field as ``pol26`` but is simpler::

            sage: pol26 = hilbert_class_polynomial(-4*26)
            sage: pol = x^6-x^5+2*x^4+x^3-2*x^2-x-1
            sage: K.<a> = NumberField(pol)
            sage: L.<b> = K.extension(x^2+26)

        Only `2` of the `j`-invariants with discriminant -104 are in
        `K`, though all are in `L`::

           sage: len(pol26.roots(K))
           2
           sage: len(pol26.roots(L))
           6

        We create an elliptic curve defined over `K` with one of the
        `j`-invariants in `K`::

            sage: j1 = pol26.roots(K)[0][0]
            sage: E = EllipticCurve(j=j1)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            False
            sage: E.has_rational_cm(L)
            True

        Over `K` the isogeny class has size `4`, with `2` curves for
        each of the `2` `K`-rational `j`-invariants::

            sage: C = E.isogeny_class(); len(C) # long time (~11s)
            4
            sage: C.matrix()                    # long time
            [ 1  2 13 26]
            [ 2  1 26 13]
            [13 26  1  2]
            [26 13  2  1]
            sage: len(Set([EE.j_invariant() for EE in C.curves]))  # long time
            2

        Over `L`, the isogeny class grows to size `6` (the class
        number)::

            sage: EL = E.change_ring(L)
            sage: CL = EL.isogeny_class(); len(CL) # long time (~21s)
            6
            sage: Set([EE.j_invariant() for EE in CL.curves]) == Set(pol26.roots(L,multiplicities=False)) # long time
            True

        In each position in the matrix of degrees, we see primes (or
        `1`).  In fact the set of degrees of cyclic isogenies from
        curve `i` to curve `j` is infinite, and is the set of all
        integers represented by one of the primitive binary quadratic
        forms of discriminant `-104`, from which we have selected a
        small prime::

            sage: CL.matrix() # long time
            [1 2 3 3 5 5]
            [2 1 5 5 3 3]
            [3 5 1 3 5 2]
            [3 5 3 1 2 5]
            [5 3 5 2 1 3]
            [5 3 2 5 3 1]

        To see the array of binary quadratic forms::

            sage: CL.qf_matrix()  # long time
            [[[1], [2, 0, 13], [3, -2, 9], [3, -2, 9], [5, -4, 6], [5, -4, 6]],
            [[2, 0, 13], [1], [5, -4, 6], [5, -4, 6], [3, -2, 9], [3, -2, 9]],
            [[3, -2, 9], [5, -4, 6], [1], [3, -2, 9], [5, -4, 6], [2, 0, 13]],
            [[3, -2, 9], [5, -4, 6], [3, -2, 9], [1], [2, 0, 13], [5, -4, 6]],
            [[5, -4, 6], [3, -2, 9], [5, -4, 6], [2, 0, 13], [1], [3, -2, 9]],
            [[5, -4, 6], [3, -2, 9], [2, 0, 13], [5, -4, 6], [3, -2, 9], [1]]]

        As in the non-CM case, the isogeny class may be visualized by
        obtaining its graph and plotting it.  Since there are more
        edges than in the non-CM case, it may be preferable to omit
        the edge_labels::

            sage: G = C.graph()
            sage: G.show(edge_labels=False) # long time

        It is possible to display a 3-dimensional plot, with colours
        to represent the different edge labels, in a form which can be
        rotated!::

            sage: G.show3d(color_by_label=True) # long time
        """
        try:
            return self._isoclass
        except AttributeError:
            from sage.schemes.elliptic_curves.isogeny_class import IsogenyClass_EC_NumberField
            self._isoclass = IsogenyClass_EC_NumberField(self)
            return self._isoclass

    def isogenies_prime_degree(self, l=None):
        r"""
        Returns a list of `\ell`-isogenies from self, where `\ell` is a
        prime.

        INPUT:

        - ``l`` -- either None or a prime or a list of primes.

        OUTPUT:

        (list) `\ell`-isogenies for the given `\ell` or if `\ell` is None, all
        isogenies of prime degree (see below for the CM case).

        .. note::

           Over `\QQ`, the codomains of the isogenies returned are
           standard minimal models.  Over other number fields they are
           global minimal models if these exist, otherwise models
           which are minimal at all but one prime.

        .. note::

           For curves with rational CM, isogenies of primes degree
           exist for infinitely many primes `\ell`, though there are
           only finitely many isogenous curves up to isomoprhism.  The
           list returned only includes one isogeny of prime degree for
           each codomain.

        EXAMPLES::

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve(K, [0,0,0,0,1])
            sage: isogs = E.isogenies_prime_degree()
            sage: [phi.degree() for phi in isogs]
            [2, 3]

            sage: pol = PolynomialRing(QQ,'x')([1,-3,5,-5,5,-3,1])
            sage: L.<a> = NumberField(pol)
            sage: js = hilbert_class_polynomial(-23).roots(L,multiplicities=False); len(js)
            3
            sage: E = EllipticCurve(j=js[0])
            sage: len(E.isogenies_prime_degree())
            3

        TESTS::

            sage: E.isogenies_prime_degree(4)
            Traceback (most recent call last):
            ...
            ValueError: 4 is not prime.

        """
        from isogeny_small_degree import isogenies_prime_degree

        if l is not None and not isinstance(l, list):
            try:
                l = ZZ(l)
            except TypeError:
                raise ValueError("%s is not a prime integer" % l)
            try:
                if l.is_prime(proof=False):
                    return isogenies_prime_degree(self, l)
                else:
                    raise ValueError("%s is not prime." % l)
            except AttributeError:
                raise ValueError("%s is not prime." % l)

        if l is None:
            from isogeny_class import possible_isogeny_degrees
            L = possible_isogeny_degrees(self)
            return self.isogenies_prime_degree(L)

        isogs = sum([self.isogenies_prime_degree(p) for p in l],[])

        if self.has_rational_cm():
            # eliminate any endomorphisms and repeated codomains
            isogs = [phi for phi in isogs if not self.is_isomorphic(phi.codomain())]
            codoms = [phi.codomain() for phi in isogs]
            isogs = [phi for i, phi in enumerate(isogs)
                     if not any([E.is_isomorphic(codoms[i])
                                 for E in codoms[:i]])]
        return isogs

    def is_isogenous(self, other, proof=True, maxnorm=100):
        """
        Returns whether or not self is isogenous to other.

        INPUT:

        - ``other`` -- another elliptic curve.

        - ``proof`` (default True) -- If ``False``, the function will
          return ``True`` whenever the two curves have the same
          conductor and are isogenous modulo `p` for all primes `p` of
          norm up to ``maxnorm``.  If ``True``, the function returns
          False when the previous condition does not hold, and if it
          does hold we compute the complete isogeny class to see if
          the curves are indeed isogenous.

        - ``maxnorm`` (integer, default 100) -- The maximum norm of
          primes `p` for which isogeny modulo `p` will be checked.

        OUTPUT:

        (bool) True if there is an isogeny from curve ``self`` to
        curve ``other``.

        EXAMPLES::

            sage: x = polygen(QQ, 'x')
            sage: F = NumberField(x^2 -2, 's'); F
            Number Field in s with defining polynomial x^2 - 2
            sage: E1 = EllipticCurve(F, [7,8])
            sage: E2 = EllipticCurve(F, [0,5,0,1,0])
            sage: E3 = EllipticCurve(F, [0,-10,0,21,0])
            sage: E1.is_isogenous(E2)
            False
            sage: E1.is_isogenous(E1)
            True
            sage: E2.is_isogenous(E2)
            True
            sage: E2.is_isogenous(E1)
            False
            sage: E2.is_isogenous(E3)
            True

        ::

            sage: x = polygen(QQ, 'x')
            sage: F = NumberField(x^2 -2, 's'); F
            Number Field in s with defining polynomial x^2 - 2
            sage: E = EllipticCurve('14a1')
            sage: EE = EllipticCurve('14a2')
            sage: E1 = E.change_ring(F)
            sage: E2 = EE.change_ring(F)
            sage: E1.is_isogenous(E2)
            True

        ::

            sage: x = polygen(QQ, 'x')
            sage: F = NumberField(x^2 -2, 's'); F
            Number Field in s with defining polynomial x^2 - 2
            sage: k.<a> = NumberField(x^3+7)
            sage: E = EllipticCurve(F, [7,8])
            sage: EE = EllipticCurve(k, [2, 2])
            sage: E.is_isogenous(EE)
            Traceback (most recent call last):
            ...
            ValueError: Second argument must be defined over the same number field.

        Some examples from Cremona's 1981 tables::

            sage: K.<i> = QuadraticField(-1)
            sage: E1 = EllipticCurve([i + 1, 0, 1, -240*i - 400, -2869*i - 2627])
            sage: E1.conductor()
            Fractional ideal (-4*i - 7)
            sage: E2 = EllipticCurve([1+i,0,1,0,0])
            sage: E2.conductor()
            Fractional ideal (-4*i - 7)
            sage: E1.is_isogenous(E2) # slower (~500ms)
            True
            sage: E1.is_isogenous(E2, proof=False) # faster  (~170ms)
            True

        In this case E1 and E2 are in fact 9-isogenous, as may be
        deduced from the following::

            sage: E3 = EllipticCurve([i + 1, 0, 1, -5*i - 5, -2*i - 5])
            sage: E3.is_isogenous(E1)
            True
            sage: E3.is_isogenous(E2)
            True
            sage: E1.isogeny_degree(E2)
            9

        TESTS:

        Check that :trac:`15890` is fixed::

            sage: K.<s> = QuadraticField(229)
            sage: c4 = 2173 - 235*(1 - s)/2
            sage: c6 = -124369 + 15988*(1 - s)/2
            sage: c4c = 2173 - 235*(1 + s)/2
            sage: c6c = -124369 + 15988*(1 + s)/2
            sage: E = EllipticCurve_from_c4c6(c4, c6)
            sage: Ec = EllipticCurve_from_c4c6(c4c, c6c)
            sage: E.is_isogenous(Ec)
            True

        Check that :trac:`17295` is fixed::

            sage: k.<s> = QuadraticField(2)
            sage: K.<b> = k.extension(x^2 - 3)
            sage: E = EllipticCurve(k, [-3*s*(4 + 5*s), 2*s*(2 + 14*s + 11*s^2)])
            sage: Ec = EllipticCurve(k, [3*s*(4 - 5*s), -2*s*(2 - 14*s + 11*s^2)])
            sage: EK = E.base_extend(K)
            sage: EcK = Ec.base_extend(K)
            sage: EK.is_isogenous(EcK)      # long time (about 3.5 s)
            True

        """
        if not is_EllipticCurve(other):
            raise ValueError("Second argument is not an Elliptic Curve.")
        if self.is_isomorphic(other):
            return True
        K = self.base_field()
        if K != other.base_field():
            raise ValueError("Second argument must be defined over the same number field.")

        E1 = self.integral_model()
        E2 = other.integral_model()
        N = E1.conductor()
        if N != E2.conductor():
            return False

        PI = K.primes_of_degree_one_iter()
        while True:
            P = next(PI)
            if P.absolute_norm() > maxnorm: break
            if not P.divides(N):
                if E1.reduction(P).cardinality() != E2.reduction(P).cardinality():
                    return False

        if not proof:
            return True

        #  We first try the easiest cases: primes for which X_0(l) has genus 0:

        for l in [2,3,5,7,13]:
            if any([E2.is_isomorphic(f.codomain()) for f in E1.isogenies_prime_degree(l)]):
                return True

        #  Next we try the primes for which X_0^+(l) has genus 0 for
        #  which isogeny-finding is faster than in general:

        from isogeny_small_degree import hyperelliptic_primes
        for l in hyperelliptic_primes:
            if any([E2.is_isomorphic(f.codomain()) for f in E1.isogenies_prime_degree(l)]):
                return True

        # Next we try looking modulo some more primes:

        while True:
            if P.absolute_norm() > 10*maxnorm: break
            if not P.divides(N):
                OP = K.residue_field(P)
                if E1.change_ring(OP).cardinality() != E2.change_ring(OP).cardinality():
                    return False
            P = next(PI)

        # Finally we compute the full isogeny class of E1 and check if
        # E2 is isomorphic to any curve in the class:

        return any([E2.is_isomorphic(E3) for E3 in E1.isogeny_class().curves])

        raise NotImplementedError("Curves appear to be isogenous (same conductor, isogenous modulo all primes of norm up to %s), but no isogeny has been constructed." % (10*maxnorm))

    def isogeny_degree(self, other):
        """
        Returns the minimal degree of an isogeny between self and
        other, or 0 if no isogeny exists.

        INPUT:

        - ``other`` -- another elliptic curve.

        OUTPUT:

        (int) The degree of an isogeny from ``self`` to ``other``, or 0.

        EXAMPLES::

            sage: x = QQ['x'].0
            sage: F = NumberField(x^2 -2, 's'); F
            Number Field in s with defining polynomial x^2 - 2
            sage: E = EllipticCurve('14a1')
            sage: EE = EllipticCurve('14a2')
            sage: E1 = E.change_ring(F)
            sage: E2 = EE.change_ring(F)
            sage: E1.isogeny_degree(E2)
            2
            sage: E2.isogeny_degree(E2)
            1
            sage: E5 = EllipticCurve('14a5').change_ring(F)
            sage: E1.isogeny_degree(E5)
            6

            sage: E = EllipticCurve('11a1')
            sage: [E2.label() for E2 in cremona_curves([11..20]) if E.isogeny_degree(E2)]
            ['11a1', '11a2', '11a3']

            sage: K.<i> = QuadraticField(-1)
            sage: E = EllipticCurve([1+i, -i, i, 1, 0])
            sage: C = E.isogeny_class()
            sage: [E.isogeny_degree(F) for F in C]
            [2, 6, 18, 9, 3, 1]
        """
        # First deal with some easy cases:
        if self.conductor() != other.conductor():
            return Integer(0)

        if self.is_isomorphic(other):
            return Integer(1)

        C = self.isogeny_class()
        i = C.index(self) # may not be 0 since curces are sorted
        try:
            return C.matrix()[i][C.index(other)]
        except ValueError:
            return ZZ(0)

    def lll_reduce(self, points, height_matrix=None, precision=None):
        """
        Returns an LLL-reduced basis from a given basis, with transform
        matrix.

        INPUT:

        - ``points`` - a list of points on this elliptic
          curve, which should be independent.

        - ``height_matrix`` - the height-pairing matrix of
          the points, or ``None``. If ``None``, it will be computed.

        - ``precision`` - number of bits of precision of intermediate
          computations (default: ``None``, for default RealField
          precision; ignored if ``height_matrix`` is supplied)

        OUTPUT: A tuple (newpoints, U) where U is a unimodular integer
        matrix, new_points is the transform of points by U, such that
        new_points has LLL-reduced height pairing matrix

        .. note::

            If the input points are not independent, the output
            depends on the undocumented behaviour of PARI's
            ``qflllgram()`` function when applied to a gram matrix which
            is not positive definite.

        EXAMPLES:

        Some examples over `\QQ`::

            sage: E = EllipticCurve([0, 1, 1, -2, 42])
            sage: Pi = E.gens(); Pi
            [(-4 : 1 : 1), (-3 : 5 : 1), (-11/4 : 43/8 : 1), (-2 : 6 : 1)]
            sage: Qi, U = E.lll_reduce(Pi)
            sage: all(sum(U[i,j]*Pi[i] for i in range(4)) == Qi[j] for j in range(4))
            True
            sage: sorted(Qi)
            [(-4 : 1 : 1), (-3 : 5 : 1), (-2 : 6 : 1), (0 : 6 : 1)]
            sage: U.det()
            1
            sage: E.regulator_of_points(Pi)
            4.59088036960573
            sage: E.regulator_of_points(Qi)
            4.59088036960574

        ::

            sage: E = EllipticCurve([1,0,1,-120039822036992245303534619191166796374,504224992484910670010801799168082726759443756222911415116])
            sage: xi = [2005024558054813068,\
            -4690836759490453344,\
            4700156326649806635,\
            6785546256295273860,\
            6823803569166584943,\
            7788809602110240789,\
            27385442304350994620556,\
            54284682060285253719/4,\
            -94200235260395075139/25,\
            -3463661055331841724647/576,\
            -6684065934033506970637/676,\
            -956077386192640344198/2209,\
            -27067471797013364392578/2809,\
            -25538866857137199063309/3721,\
            -1026325011760259051894331/108241,\
            9351361230729481250627334/1366561,\
            10100878635879432897339615/1423249,\
            11499655868211022625340735/17522596,\
            110352253665081002517811734/21353641,\
            414280096426033094143668538257/285204544,\
            36101712290699828042930087436/4098432361,\
            45442463408503524215460183165/5424617104,\
            983886013344700707678587482584/141566320009,\
            1124614335716851053281176544216033/152487126016]
            sage: points = [E.lift_x(x) for x in xi]
            sage: newpoints, U = E.lll_reduce(points)  # long time (35s on sage.math, 2011)
            sage: [P[0] for P in newpoints]            # long time
            [6823803569166584943, 5949539878899294213, 2005024558054813068, 5864879778877955778, 23955263915878682727/4, 5922188321411938518, 5286988283823825378, 175620639884534615751/25, -11451575907286171572, 3502708072571012181, 1500143935183238709184/225, 27180522378120223419/4, -5811874164190604461581/625, 26807786527159569093, 7404442636649562303, 475656155255883588, 265757454726766017891/49, 7272142121019825303, 50628679173833693415/4, 6951643522366348968, 6842515151518070703, 111593750389650846885/16, 2607467890531740394315/9, -1829928525835506297]

        An example to show the explicit use of the height pairing matrix::

            sage: E = EllipticCurve([0, 1, 1, -2, 42])
            sage: Pi = E.gens()
            sage: H = E.height_pairing_matrix(Pi,3)
            sage: E.lll_reduce(Pi,height_matrix=H)
            (
                                                                      [1 0 0 1]
                                                                      [0 1 0 1]
                                                                      [0 0 0 1]
            [(-4 : 1 : 1), (-3 : 5 : 1), (-2 : 6 : 1), (1 : -7 : 1)], [0 0 1 1]
            )

        Some examples over number fields (see :trac:`9411`)::

            sage: K.<a> = QuadraticField(-23, 'a')
            sage: E = EllipticCurve(K, '37')
            sage: E.lll_reduce(E.gens())
            (
                                                    [ 1 -1]
            [(0 : 0 : 1), (-2 : -1/2*a - 1/2 : 1)], [ 0  1]
            )

        ::

            sage: K.<a> = QuadraticField(-5)
            sage: E = EllipticCurve(K,[0,a])
            sage: points = [E.point([-211/841*a - 6044/841,-209584/24389*a + 53634/24389]),E.point([-17/18*a - 1/9, -109/108*a - 277/108]) ]
            sage: E.lll_reduce(points)
            (
            [(-a + 4 : -3*a + 7 : 1), (-17/18*a - 1/9 : 109/108*a + 277/108 : 1)],
            [ 1  0]
            [ 1 -1]
            )
        """
        r = len(points)
        if height_matrix is None:
            height_matrix = self.height_pairing_matrix(points, precision)
        U = height_matrix._pari_().lllgram().python()
        new_points = [sum([U[j, i]*points[j] for j in range(r)])
                      for i in range(r)]
        return new_points, U

    def galois_representation(self):
        r"""
        The compatible family of the Galois representation
        attached to this elliptic curve.

        Given an elliptic curve `E` over a number field `K`
        and a rational prime number `p`, the `p^n`-torsion
        `E[p^n]` points of `E` is a representation of the
        absolute Galois group of `K`. As `n` varies
        we obtain the Tate module `T_p E` which is a
        a representation of `G_K` on a free `\ZZ_p`-module
        of rank `2`. As `p` varies the representations
        are compatible.

        EXAMPLES::

            sage: K = NumberField(x**2 + 1, 'a')
            sage: E = EllipticCurve('11a1').change_ring(K)
            sage: rho = E.galois_representation()
            sage: rho
            Compatible family of Galois representations associated to the Elliptic Curve defined by y^2 + y = x^3 + (-1)*x^2 + (-10)*x + (-20) over Number Field in a with defining polynomial x^2 + 1
            sage: rho.is_surjective(3)
            True
            sage: rho.is_surjective(5)  # long time (4s on sage.math, 2014)
            False
            sage: rho.non_surjective()
            [5]
        """
        return gal_reps_number_field.GaloisRepresentation(self)

    def cm_discriminant(self):
        """
        Returns the CM discriminant of the `j`-invariant of this curve.

        OUTPUT:

        An integer `D` which is either `0` if this curve `E` does not
        have Complex Multiplication) (CM), or an imaginary quadratic
        discriminant if `j(E)` is the `j`-invariant of the order with
        discriminant `D`.

        .. note::

           If `E` has CM but the discriminant `D` is not a square in
           the base field `K` then the extra endomorphisms will not be
           defined over `K`.  See also :meth:`has_rational_cm`.

        EXAMPLES::

            sage: EllipticCurve(j=0).cm_discriminant()
            -3
            sage: EllipticCurve(j=1).cm_discriminant()
            Traceback (most recent call last):
            ...
            ValueError: Elliptic Curve defined by y^2 + x*y = x^3 + 36*x + 3455 over Rational Field does not have CM
            sage: EllipticCurve(j=1728).cm_discriminant()
            -4
            sage: EllipticCurve(j=8000).cm_discriminant()
            -8
            sage: K.<a> = QuadraticField(5)
            sage: EllipticCurve(j=282880*a + 632000).cm_discriminant()
            -20
            sage: K.<a> = NumberField(x^3 - 2)
            sage: EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000).cm_discriminant()
            -108
        """
        try:
            D = self._CMD
        except AttributeError:
            pass
        else:
            if D:
                return D
            else:
                raise ValueError("%s does not have CM"%self)

        from sage.schemes.elliptic_curves.cm import is_cm_j_invariant
        flag, df =  is_cm_j_invariant(self.j_invariant())
        if flag:
            d, f = df
            self._CMD = d*f**2
            return self._CMD
        self._CMD = 0 # special cached value to indicate no CM
        raise ValueError("%s does not have CM"%self)

    def has_cm(self):
        """
        Returns whether or not this curve has a CM `j`-invariant.

        OUTPUT:

        ``True`` if this curve has CM over the algebraic closure
        of the base field, otherwise ``False``.  See also
        :meth:`cm_discriminant()` and :meth:`has_rational_cm`.

        .. note::

           Even if `E` has CM in this sense (that its `j`-invariant is
           a CM `j`-invariant), since the associated negative
           discriminant `D` is not a square in the base field `K`, the
           extra endomorphisms will not be defined over `K`.  See also
           the method :meth:`has_rational_cm` which tests whether `E`
           has extra endomorphisms defined over `K` or a given
           extension of `K`.

        EXAMPLES::

            sage: EllipticCurve(j=0).has_cm()
            True
            sage: EllipticCurve(j=1).has_cm()
            False
            sage: EllipticCurve(j=1728).has_cm()
            True
            sage: EllipticCurve(j=8000).has_cm()
            True
            sage: K.<a> = QuadraticField(5)
            sage: EllipticCurve(j=282880*a + 632000).has_cm()
            True
            sage: K.<a> = NumberField(x^3 - 2)
            sage: EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000).has_cm()
            True
        """
        try:
            D = self.cm_discriminant()
        except ValueError:
            return False
        return True

    def has_rational_cm(self, field=None):
        """
        Returns whether or not this curve has CM defined over its
        base field or a given extension.

        INPUT:

        - ``field`` -- a field, which should be an extension of the
          base field of the curve.  If ``field`` is ``None`` (the
          default), it is taken to be the base field of the curve.

        OUTPUT:

        ``True`` if the ring of endomorphisms of this curve over
        the given field is larger than `\ZZ`; otherwise ``False``.
        See also :meth:`cm_discriminant()` and :meth:`has_cm`.

        .. note::

           If `E` has CM but the discriminant `D` is not a square in
           the given field `K` then the extra endomorphisms will not
           be defined over `K`, and this function will return
           ``False``.  See also :meth:`has_cm`.  To obtain the CM
           discriminant, use :meth:`cm_discriminant()`.

        EXAMPLES::

            sage: E = EllipticCurve(j=0)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            False
            sage: D = E.cm_discriminant(); D
            -3
            sage: E.has_rational_cm(QuadraticField(D))
            True

            sage: E = EllipticCurve(j=1728)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            False
            sage: D = E.cm_discriminant(); D
            -4
            sage: E.has_rational_cm(QuadraticField(D))
            True

        Higher degree examples::

            sage: K.<a> = QuadraticField(5)
            sage: E = EllipticCurve(j=282880*a + 632000)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            False
            sage: E.cm_discriminant()
            -20
            sage: E.has_rational_cm(K.extension(x^2+5,'b'))
            True

        An error is raised if a field is given which is not an extension of the base field::

            sage: E.has_rational_cm(QuadraticField(-20))
            Traceback (most recent call last):
            ...
            ValueError: Error in has_rational_cm: Number Field in a with defining polynomial x^2 + 20 is not an extension field of Number Field in a with defining polynomial x^2 - 5

            sage: K.<a> = NumberField(x^3 - 2)
            sage: E = EllipticCurve(j=31710790944000*a^2 + 39953093016000*a + 50337742902000)
            sage: E.has_cm()
            True
            sage: E.has_rational_cm()
            False
            sage: D = E.cm_discriminant(); D
            -108
            sage: E.has_rational_cm(K.extension(x^2+108,'b'))
            True
        """
        try:
            D = self.cm_discriminant()
        except ValueError:
            return False
        if field is None:
            return self.base_field()(D).is_square()
        if self.base_field().embeddings(field):
            D = field(D)
            return D.is_square()
        raise ValueError("Error in has_rational_cm: %s is not an extension field of %s"
                         % (field,self.base_field()))
