Subsections

14. Elliptic curves


14.1 conductor

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


14.2 $ j$ -invariant

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.weierstrass_model()
      Elliptic Curve defined by y^2  = x^3 - 13392*x - 1080432 over Rational Field
sage: E.discriminant()
      -161051


14.3 The $ GF(q)$ -rational points on E

Given an elliptic curve defined over $ {\mathbb{F}}= GF(q)$ , SAGE can compute its set or $ {\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(); G
(Multiplicative Abelian Group isomorphic to C5, ((1 : 0 : 1),))
sage: G[0].permutation_group()
Permutation Group with generators [(1,2,3,4,5)]

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