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18.783 | Fall 2025 | Undergraduate

# Elliptic Curves

### Sage/CoCalc

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.)

### LMFDB

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 500,000 that includes 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

##### Instructor

- Andrew V. Sutherland

##### Departments

- Mathematics

##### As Taught In

Fall 2025

##### Level

Undergraduate

##### Topics

- Engineering
  - Computer Science
- Mathematics
  - Algebra and Number Theory
  - Computation

##### Learning Resource Types

notes Lecture Notes
menu_book Open Textbooks
assignment Problem Sets
auto_stories Readings

Browse Resources

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