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# Oxford University Elliptic Curves Research Project

### Computing the average rank of elliptic curves in ordered families

Supervisor: Dr Jennifer Balakrishnan

An elliptic curve $E$ is the set of points $(x, y)$ satisfying an equation of the form $y^2 = x^3 + ax + b$, together with a special point “at infinity”. When the coefficients $a$, $b$ are rational numbers, it is of interest to study the set of all rational points $(x, y)$ satisfying the equation. It turns out that the rational points on $E$ form a finitely generated abelian group, and an important invariant attached to $E$ is the rank of this group. During the project we explored the “average” rank of various elliptic curves in certain families by using Sage to build a database of millions of elliptic curves, then computing various important number theoretic invariants attached to these curves.