Elliptic curves over GF(p)?

[Alasdair McAndrew <amca01@gmail.com>]

Hello,

I want to do some simple computation on elliptic curves of the form y^2 = x^3 + ax +b (mod p), where p is prime (of at least 32 bits) .  My needs are simple: define such a curve, find its cardinality, find a generator (if the cardinality is prime) or a point of high order, and in general perform arithmetic on the curve: addition, order, etc.  Can libpari manage all of this?  The manual is fairly sparse on elliptic curves, and it's not clear to me, as a beginner, whether Pari supports elliptic curves over finite fields GF(p). 

Thanks very much!

cheers,

Alasdair