On the torsion of rational elliptic curves over sextic fields

Abstract

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G=E(\mathbb{Q}{)}_{\mathrm{tors}}$ we study what groups (up to isomorphism) can occur as the torsion subgroup of $E$ base-extended to $K$, a degree 6 extension of $\mathbb{Q}$. We also determine which groups $H=E(K{)}_{\mathrm{tors}}$ can occur infinitely often and which ones occur for only finitely many curves. This article is a first step towards a complete classification of torsion growth of over sextic fields.

To appear in Mathematics of Computation