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# Projective plane conics over \(\QQ\)

AUTHORS:

- Marco Streng (2010-07-20)
- Nick Alexander (2008-01-08)

class `sage.schemes.plane_conics.con_rational_field.``ProjectiveConic_rational_field`(A, f)

  Bases: `sage.schemes.plane_conics.con_number_field.ProjectiveConic_number_field`

  Create a projective plane conic curve over \(\QQ\). See `Conic` for full documentation.

  EXAMPLES:

      sage: P.<X, Y, Z> = QQ[]
      sage: Conic(X^2 + Y^2 - 3*Z^2)
      Projective Conic Curve over Rational Field defined by X^2 + Y^2 - 3*Z^2

  TESTS:

      sage: Conic([2, 1, -1])._test_pickling()

  `has_rational_point`(point=False, obstruction=False, algorithm='default', read_cache=True)

    Returns True if and only if `self` has a point defined over \(\QQ\).

    If `point` and `obstruction` are both False (default), then the output is a boolean `out` saying whether `self` has a rational point.

    If `point` or `obstruction` is True, then the output is a pair `(out, S)`, where `out` is as above and the following holds:

    > - if `point` is True and `self` has a rational point, then `S` is a rational point,
    > - if `obstruction` is True and `self` has no rational point, then `S` is a prime such that no rational point exists over the completion at `S` or \(-1\) if no point exists over \(\RR\).

    Points and obstructions are cached, whenever they are found. Cached information is used if and only if `read_cache` is True.

    ALGORITHM:

    The parameter `algorithm` specifies the algorithm to be used:

    > - `'qfsolve'` – Use Denis Simon’s GP script `qfsolve` (see `sage.quadratic_forms.qfsolve.qfsolve`)
    > - `'rnfisnorm'` – Use PARI’s function rnfisnorm (cannot be combined with `obstruction = True`)
    > - `'local'` – Check if a local solution exists for all primes and infinite places of \(\QQ\) and apply the Hasse principle (cannot be combined with `point = True`)
    > - `'default'` – Use `'qfsolve'`
    > - `'magma'` (requires Magma to be installed) – delegates the task to the Magma computer algebra system.

    EXAMPLES:

        sage: C = Conic(QQ, [1, 2, -3])
        sage: C.has_rational_point(point = True)
        (True, (1 : 1 : 1))
        sage: D = Conic(QQ, [1, 3, -5])
        sage: D.has_rational_point(point = True)
        (False, 3)
        sage: P.<X,Y,Z> = QQ[]
        sage: E = Curve(X^2 + Y^2 + Z^2); E
        Projective Conic Curve over Rational Field defined by X^2 + Y^2 + Z^2
        sage: E.has_rational_point(obstruction = True)
        (False, -1)

    The following would not terminate quickly with `algorithm = 'rnfisnorm'`

        sage: C = Conic(QQ, [1, 113922743, -310146482690273725409])
        sage: C.has_rational_point(point = True)
        (True, (-76842858034579/5424 : -5316144401/5424 : 1))
        sage: C.has_rational_point(algorithm = 'local', read_cache = False)
        True
        sage: C.has_rational_point(point=True, algorithm='magma', read_cache=False) # optional - magma
        (True, (30106379962113/7913 : 12747947692/7913 : 1))

    TESTS:

    Create a bunch of conics over \(\QQ\), check if `has_rational_point` runs without errors and returns consistent answers for all algorithms. Check if all points returned are valid.

        sage: l = Sequence(cartesian_product_iterator([[-1, 0, 1] for i in range(6)]))
        sage: c = [Conic(QQ, a) for a in l if a != [0,0,0] and a != (0,0,0,0,0,0)]
        sage: d = []
        sage: d = [[C]+[C.has_rational_point(algorithm = algorithm, read_cache = False, obstruction = (algorithm != 'rnfisnorm'), point = (algorithm != 'local')) for algorithm in ['local', 'qfsolve', 'rnfisnorm']] for C in c[::10]] # long time: 7 seconds
        sage: assert all([e[1][0] == e[2][0] and e[1][0] == e[3][0] for e in d])
        sage: assert all([e[0].defining_polynomial()(Sequence(e[i][1])) == 0 for e in d for i in [2,3] if e[1][0]])

  `is_locally_solvable`(p)

    Returns True if and only if `self` has a solution over the \(p\)-adic numbers. Here \(p\) is a prime number or equals \(-1\), infinity, or \(\RR\) to denote the infinite place.

    EXAMPLES:

        sage: C = Conic(QQ, [1,2,3])
        sage: C.is_locally_solvable(-1)
        False
        sage: C.is_locally_solvable(2)
        False
        sage: C.is_locally_solvable(3)
        True
        sage: C.is_locally_solvable(QQ.hom(RR))
        False
        sage: D = Conic(QQ, [1, 2, -3])
        sage: D.is_locally_solvable(infinity)
        True
        sage: D.is_locally_solvable(RR)
        True

  `local_obstructions`(finite=True, infinite=True, read_cache=True)

    Returns the sequence of finite primes and/or infinite places such that self is locally solvable at those primes and places.

    The infinite place is denoted \(-1\).

    The parameters `finite` and `infinite` (both True by default) are used to specify whether to look at finite and/or infinite places. Note that `finite = True` involves factorization of the determinant of `self`, hence may be slow.

    Local obstructions are cached. The parameter `read_cache` specifies whether to look at the cache before computing anything.

    EXAMPLES

        sage: Conic(QQ, [1, 1, 1]).local_obstructions()
        [2, -1]
        sage: Conic(QQ, [1, 2, -3]).local_obstructions()
        []
        sage: Conic(QQ, [1, 2, 3, 4, 5, 6]).local_obstructions()
        [41, -1]

  `parametrization`(point=None, morphism=True)

    Return a parametrization \(f\) of `self` together with the inverse of \(f\).

    If `point` is specified, then that point is used for the parametrization. Otherwise, use `self.rational_point()` to find a point.

    If `morphism` is True, then \(f\) is returned in the form of a Scheme morphism. Otherwise, it is a tuple of polynomials that gives the parametrization.

    ALGORITHM:

    Uses Denis Simon’s GP script `qfparam`. See `sage.quadratic_forms.qfsolve.qfparam`.

    EXAMPLES

        sage: c = Conic([1,1,-1])
        sage: c.parametrization()
        (Scheme morphism:
          From: Projective Space of dimension 1 over Rational Field
          To:   Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
          Defn: Defined on coordinates by sending (x : y) to
                (2*x*y : -x^2 + y^2 : x^2 + y^2),
         Scheme morphism:
          From: Projective Conic Curve over Rational Field defined by x^2 + y^2 - z^2
          To:   Projective Space of dimension 1 over Rational Field
          Defn: Defined on coordinates by sending (x : y : z) to
                (1/2*x : 1/2*y + 1/2*z))

    An example with `morphism = False`

        sage: R.<x,y,z> = QQ[]
        sage: C = Curve(7*x^2 + 2*y*z + z^2)
        sage: (p, i) = C.parametrization(morphism = False); (p, i)
        ([-2*x*y, 7*x^2 + y^2, -2*y^2], [-1/2*x, -1/2*z])
        sage: C.defining_polynomial()(p)
        0
        sage: i[0](p) / i[1](p)
        x/y

    A `ValueError` is raised if `self` has no rational point

        sage: C = Conic(x^2 + 2*y^2 + z^2)
        sage: C.parametrization()
        Traceback (most recent call last):
        ...
        ValueError: Conic Projective Conic Curve over Rational Field defined by x^2 + 2*y^2 + z^2 has no rational points over Rational Field!

    A `ValueError` is raised if `self` is not smooth

        sage: C = Conic(x^2 + y^2)
        sage: C.parametrization()
        Traceback (most recent call last):
        ...
        ValueError: The conic self (=Projective Conic Curve over Rational Field defined by x^2 + y^2) is not smooth, hence does not have a parametrization.

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