14H52: Elliptic Curves
This is a fascinating area of algebraic geometry dealing with nonsingular curves of genus 1 -- in English, solutions to equations y^2 = x^3 + A x + B. It turns out to have important connections to number theory and in particular to factorization of ordinary integers (and thus to cryptography). Also, what appear to be simple Diophantine equations often lead to elliptic curves. Through Riemann surfaces it has connections to topology; through modular forms and zeta functions to analysis. Elliptic curves also played a role in the recent resolution of the conjecture known as Fermat's Last Theorem.
See also 11G05, 11G07, 14Kxx
Parent field: 14H - Algebraic Curves
Online texts on elliptic curves by James Milne and Ian Connell
If you have Maple, get Connell's APECS package, which does nearly everything anyone knows an algorithm for on an elliptic curve. Here's a link to APECS.
If you need to compute ranks of elliptic curves over the rationals, you definitely need Cremona's MWRANK. See his book or his web site for more information.