Number of points on elliptic curve

Question

If you have an elliptic curve in the form of:

y^2 = x^3 + a*x + b  (mod p)

Is there a good program to calculate the number of points on this cu

Answer 1

Have you heard of Sage?

Sage includes Pari, which is an open source package for number theory. Pari has an implementation of SEA.

From http://wstein.org/papers/2008-bordeaux/sphinx/elliptic_curves.html#schoof-elkies-atkin-point-counting:

sage: k = GF(next_prime(10^20))
sage: E = EllipticCurve(k.random_element())
sage: E.cardinality()                   # less than a second
100000000005466254167