Abstract

In 2016, Balakrishnan, Ho, Kaplan, Spicer, Stein and Weigandt produced a database of elliptic curves over $\mathbb{Q}$ ordered by height in which they computed the rank, the size of the $2$-Selmer group, and other arithmetic invariants. They observed that after a certain point, the average rank seemed to decrease as the height increased. Here we consider the family of elliptic curves over $\mathbb{Q}$ whose rational torsion subgroup is isomorphic to $\mathbb{Z}∕2\mathbb{Z}\times \mathbb{Z}∕8\mathbb{Z}$. Conditional on GRH and BSD, we compute the rank of $92\%$ of the $202,461$ curves with parameter height less than $10^3$. We also compute the size of the $2$-Selmer group and the Tamagawa product, and prove that their averages tend to infinity for this family.

Keywords

Mathematical Subject Classification 2010

Milestones

Authors