### Navigation

- index
- modules |
- next |
- previous |
- Sage Constructions v6.4.1 »

# Elliptic curves

## Conductor

How do you compute the conductor of an elliptic curve (over \(\QQ\)) in Sage?

Once you define an elliptic curve \(E\) in Sage, using the `EllipticCurve` command, the conductor is one of several “methods” associated to \(E\). Here is an example of the syntax (borrowed from section 2.4 “Modular forms” in the tutorial):

    sage: E = EllipticCurve([1,2,3,4,5])
    sage: E
    Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over
    Rational Field
    sage: E.conductor()
    10351

## \(j\)-invariant

How do you compute the \(j\)-invariant of an elliptic curve in Sage?

Other methods associated to the `EllipticCurve` class are `j_invariant`, `discriminant`, and `weierstrass_model`. Here is an example of their syntax.

    sage: E = EllipticCurve([0, -1, 1, -10, -20])
    sage: E
    Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
    sage: E.j_invariant()
    -122023936/161051
    sage: E.short_weierstrass_model()
    Elliptic Curve defined by y^2  = x^3 - 13392*x - 1080432 over Rational Field
    sage: E.discriminant()
    -161051
    sage: E = EllipticCurve(GF(5),[0, -1, 1, -10, -20])
    sage: E.short_weierstrass_model()
    Elliptic Curve defined by y^2  = x^3 + 3*x + 3 over Finite Field of size 5
    sage: E.j_invariant()
    4

## The \(GF(q)\)-rational points on E

How do you compute the number of points of an elliptic curve over a finite field?

Given an elliptic curve defined over \(\mathbb{F} = GF(q)\), Sage can compute its set of \(\mathbb{F}\)-rational points

    sage: E = EllipticCurve(GF(5),[0, -1, 1, -10, -20])
    sage: E
    Elliptic Curve defined by y^2 + y = x^3 + 4*x^2 over Finite Field of size 5
    sage: E.points()
    [(0 : 0 : 1), (0 : 1 : 0), (0 : 4 : 1), (1 : 0 : 1), (1 : 4 : 1)]
    sage: E.cardinality()
    5
    sage: G = E.abelian_group()
    sage: G
    Additive abelian group isomorphic to Z/5 embedded in Abelian group of points on Elliptic Curve defined by y^2 + y = x^3 + 4*x^2 over Finite Field of size 5
    sage: G.permutation_group()
    Permutation Group with generators [(1,2,3,4,5)]

## Modular form associated to an elliptic curve over \(\QQ\)

Let \(E\) be a “nice” elliptic curve whose equation has integer coefficients, let \(N\) be the conductor of \(E\) and, for each \(n\), let \(a_n\) be the number appearing in the Hasse-Weil \(L\)-function of \(E\). The Taniyama-Shimura conjecture (proven by Wiles) states that there exists a modular form of weight two and level \(N\) which is an eigenform under the Hecke operators and has a Fourier series \(\sum_{n = 0}^\infty a_n q^n\). Sage can compute the sequence \(a_n\) associated to \(E\). Here is an example.

    sage: E = EllipticCurve([0, -1, 1, -10, -20])
    sage: E
    Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
    sage: E.conductor()
    11
    sage: E.anlist(20)
    [0, 1, -2, -1, 2, 1, 2, -2, 0, -2, -2, 1, -2, 4, 4, -1, -4, -2, 4, 0, 2]
    sage: E.analytic_rank()
    0

### Navigation

- index
- modules |
- next |
- previous |
- Sage Constructions v6.4.1 »