Mathematica and Maple: Both are commercial, but they are very
powerful, can draw pretty pictures, and there are elliptic curve
packages available for each (apecs for Maple, and something
by Silverman for Mathematica).
PARI: Free, open source, excellent for our course, simple,
runs on Macs, MS Windows, Linux, etc.
MAGMA: Huge, non-free but nonprofit,
what I usually use for my research. I can legally give you a Linux
executable if you are registered for 124.
My Wristwatch: Perhaps the only wristwatch in the world
that can factor your social security number? :-)