Elliptic curves

Conductor

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

Once you define an elliptic curve in Sage, using the EllipticCurve command, the conductor is one of several “methods” associated to . 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

-invariant

How do you compute the -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 -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 , Sage can compute its set of -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              # random choice of generator
(Multiplicative Abelian Group isomorphic to C5, ((1 : 0 : 1),))
sage: G[0].permutation_group()
Permutation Group with generators [(1,2,3,4,5)]

Modular form associated to an elliptic curve over

Let be a “nice” elliptic curve whose equation has integer coefficients, let be the conductor of and, for each , let be the number appearing in the Hasse-Weil -function of . The Taniyama-Shimura conjecture (proven by Wiles) states that there exists a modular form of weight two and level which is an eigenform under the Hecke operators and has a Fourier series . Sage can compute the sequence associated to . 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