Torsion of rational elliptic curves over cubic fields and sporadic points on $X_1(n)$

TL;DR

This paper classifies torsion structures of rational elliptic curves over cubic fields, discovering a new torsion structure and identifying a minimal degree sporadic point on a modular curve.

Contribution

It provides a complete classification of torsion groups over cubic fields and finds a novel torsion structure, /21, linked to a minimal-degree sporadic point on X_1(21).

Findings

Identified /21 as a new torsion structure over a cubic field.

Found a sporadic point of degree 3 on X_1(21), the lowest known for such points.

Classified all possible torsion structures of rational elliptic curves over cubic fields.

Abstract

We classify the possible torsion structures of rational elliptic curves over cubic fields. Along the way we find a previously unknown torsion structure over a cubic field, $\Z /21 \Z$, which corresponds to a sporadic point on $X_1(21)$ of degree 3, which is the lowest possible degree of a sporadic point on a modular curve $X_1(n)$.