Elliptic curves

Conductor

How do you compute the conductor of an elliptic curve (over $\mathbf{Q}$) 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

Sage Live

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

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

$j$-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

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()

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

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)]

Sage Live

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

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)]

Modular form associated to an elliptic curve over $\mathbf{Q}$

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 $\sum _{n=0}^{\mathrm{\infty }}{a}_n{q}^n$. 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

Sage Live

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

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

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