# 椭圆曲线¶

## 导线¶

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


## $$j$$ -不变量¶

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


## 这个 $$GF(q)$$ -E上的有理点¶

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
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)]


## 与椭圆曲线有关的模形式 $$\QQ$$¶

$$E$$ 是一条“漂亮”的椭圆曲线，它的方程有整系数，让 $$N$$ 做…的指挥 $$E$$ 而且，对于每一个 $$n$$ ，让 $$a_n$$ 是哈斯韦尔的号码吗 $$L$$ -功能 $$E$$ . Taniyama-Shimura猜想（由Wiles证明）指出存在一个权重为2和水平的模形式 $$N$$ 它是Hecke算子下的一个特征形式，有一个Fourier级数 $$\sum_{{n = 0}}^\infty a_n q^n$$ . Sage可以计算序列 $$a_n$$ 关联到 $$E$$ . 这里有一个例子。

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