13.3 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 : 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              # random choice of generator
(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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