18.783 | Spring 2021 | Undergraduate

Elliptic Curves

Related Resources

Collaborative Calculation and Data Science

Sage is an open-source computer algebra system with extensive support for computing with elliptic curves. CoCalc is a hosted collaboration site that provides a notebook interface to Sage which you can access via any web browser. No software installation is required, but you will need to create a CoCalc account to log in (This is free of charge to students; I will cover CoCalc hosting charges related to the course.) All Sage worksheets used in lectures will be published on CoCalc, as will example solutions to problem sets. You can also download a copy of Sage to run on your own machine if you wish, or you can run Sage from the command line on any of the mathematics department servers (You will still need to use CoCalc to download code and worksheets associated to problem sets.)


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 arithmetic 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.

Course Info

As Taught In
Spring 2021
Learning Resource Types
Problem Sets
Lecture Notes
Instructor Insights