18.783 - Elliptic Curves



Sage is an open-source computer algebra system with extensive support for computing with elliptic curves. You can access Sage via SageMathCloud using any browser, no software installation is required. Both a graphical user interface and command-line interface are provided. All of the worksheets used in lectures will be published on SageMathCloud, as will example solutions to problem sets. Tutorials and many examples can be found on-line (google is your friend). You can also download a copy of Sage to run on your own machine if you wish.


The L-functions and modular forms database contains a wealth of information about elliptic curves. modular forms, and their L-functions, including a complete list of all elliptic curves over Q of conductor up to 400,000 (and counting) that incorporates isogeny graphs, images of Galois representations, rational points, and invariants related to the conjecture of Birch and Swinnerton-Dyer.

Explicit Formulas Database

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.