Sage is an open-source computer algebra system with extensive support for computing with elliptic curves. Sage provides both a command-line interface and a browser-based GUI (the Sage notebook). A Sage notebook server has been setup specifically for use in this course (click on the link above). All of the worksheets used in lectures will be published there, as will example solutions to problem sets. Tutorials and many examples can be found on-line (google is your friend). You can download a copy of Sage to run on your own machine if you wish.
Explicit formulas implementing the group operation on elliptic curves in a wide variety of forms can be found here. These formulas have been carefully optimized and verified for correctness.