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

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

Python

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

Sage Live

E = EllipticCurve([1,2,3,4,5])
E
E.conductor()

\(j\)-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

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

Python

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

Sage Live

E = EllipticCurve([0, -1, 1, -10, -20])
E
E.j_invariant()
E.short_weierstrass_model()
E.discriminant()
E = EllipticCurve(GF(5),[0, -1, 1, -10, -20])
E.short_weierstrass_model()
E.j_invariant()

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

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 : 1 : 0), (0 : 0 : 1), (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)]

Python

>>> from sage.all import *
>>> E = EllipticCurve(GF(Integer(5)),[Integer(0), -Integer(1), Integer(1), -Integer(10), -Integer(20)])
>>> E
Elliptic Curve defined by y^2 + y = x^3 + 4*x^2 over Finite Field of size 5
>>> E.points()
[(0 : 1 : 0), (0 : 0 : 1), (0 : 4 : 1), (1 : 0 : 1), (1 : 4 : 1)]
>>> E.cardinality()
5
>>> G = E.abelian_group()
>>> 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
>>> G.permutation_group()
Permutation Group with generators [(1,2,3,4,5)]

Sage Live

E = EllipticCurve(GF(5),[0, -1, 1, -10, -20])
E
E.points()
E.cardinality()
G = E.abelian_group()
G
G.permutation_group()

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

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

Python

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

Sage Live

E = EllipticCurve([0, -1, 1, -10, -20])
E
E.conductor()
E.anlist(20)
E.analytic_rank()