Torsion of elliptic curves with rational $j$-invariant defined over number fields of prime degree

TL;DR

This paper classifies the possible torsion subgroups of elliptic curves with rational j-invariant over prime degree number fields by analyzing Galois representations and quadratic twists.

Contribution

It provides a complete classification of torsion subgroups for such elliptic curves over prime degree number fields, extending previous results.

Findings

Complete list of torsion subgroups over prime degree fields

Use of Galois representations to determine torsion possibilities

Analysis of quadratic twists to refine classifications

Abstract

Let $[K:\mathbb{Q}]=p$ be a prime number and let $E/K$ be an elliptic curve with $j(E) \in \mathbb{Q}$. We determine the all possibilities for $E(K)_{tors}$. We obtain these results by studying Galois representations of $E$ and of it's quadratic twists.