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.