椭圆曲线

导子

如何在 Sage 中计算椭圆曲线（在 \(\QQ\) 上）的导子？

在 Sage 中使用 EllipticCurve 命令定义椭圆曲线 \(E\) 后， 导子就是与 \(E\) 关联的若干“方法”之一。以下是语法示例（来自教程第 2.4 节“模形式”）：

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

\(j\)-不变量

如何在 Sage 中计算椭圆曲线的 \(j\)-不变量？

与 EllipticCurve 类相关的其他方法包括 j_invariant, discriminant 和 weierstrass_model。以下是语法示例：

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

E 上的 \(GF(q)\)-有理点

如何计算有限域上椭圆曲线的点数？

给定一个定义在 \(\mathbb{F} = GF(q)\) 上的椭圆曲线， Sage 可以计算其 \(\mathbb{F}\)-有理点集。

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)]

与 \(\QQ\) 上椭圆曲线相关的模形式

设 \(E\) 是一个“良好”的椭圆曲线，其方程具有整数系数。 设 \(N\) 为 \(E\) 的导子，并且对于每个 \(n\)， 设 \(a_n\) 是出现在 \(E\) 的 Hasse-Weil \(L\)-函数中的数字。 Taniyama-Shimura 猜想（已被 Wiles 证明）表明存在一个权重为 2、级别为 \(N\) 的模形式， 它是 Hecke 算子下的特征形式，并具有傅里叶级数 \(\sum_{n = 0}^\infty a_n q^n\)。 Sage 可以计算与 \(E\) 相关的序列 \(a_n\)。以下是一个示例。

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