On the torsion of rational elliptic curves over sextic fields

TL;DR

This paper investigates how the torsion subgroup of rational elliptic curves can grow when extended to degree 6 fields, providing initial classifications of possible torsion groups and their frequency.

Contribution

It offers the first systematic study of torsion growth of rational elliptic curves over sextic fields, identifying which torsion groups can occur and their frequency.

Findings

Classified possible torsion groups over sextic fields.

Determined which torsion groups occur infinitely often.

Identified torsion groups that occur only finitely often.

Abstract

Given an elliptic curve $E/\mathbb{Q}$ with torsion subgroup $G = E(\mathbb{Q})_{\rm tors}$ 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 $H = E(K)_{\rm tors}$ 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.