Computing the average rank of elliptic curves in ordered families

Supervisor: 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.