Torsion of elliptic curves with rational $j$-invariant over quartic number fields

TL;DR

This paper classifies the possible torsion subgroup structures of elliptic curves with rational j-invariant over quartic number fields, expanding understanding of elliptic curve torsion phenomena in higher degree fields.

Contribution

It provides a complete classification of torsion subgroups for elliptic curves with rational j-invariant over quartic fields, a previously unresolved case.

Findings

Identifies all possible torsion subgroup structures over quartic fields.

Establishes constraints on torsion growth in these fields.

Complements existing classifications over lower degree fields.

Abstract

Let $E$ be an elliptic curve, defined over a quartic extension $K$ of $\mathbb{Q}$, with $j(E) \in \mathbb{Q}$. In this paper, we classify the possible torsion subgroup structures $E(K)_{\text{tors}}$.