# 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

>>> 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


## $$j$$-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

>>> 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


## 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 : 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)]

>>> 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)]


## 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

>>> 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