Degrees of isogenies over prime degree number fields of non-CM elliptic curves with rational $j$-invariant

TL;DR

This paper classifies all possible degrees of cyclic isogenies for non-CM elliptic curves with rational j-invariant over prime degree number fields, completing the case for odd primes.

Contribution

It provides a complete classification of isogeny degrees over prime degree fields for non-CM elliptic curves with rational j-invariant, extending previous results for p=2.

Findings

All possible degrees of cyclic isogenies over prime degree fields are determined.

The classification is complete for odd prime degrees, filling a gap in the literature.

The results generalize known classifications for p=2.

Abstract

We determine all possible degrees of cyclic isogenies of non-CM elliptic curves with rational $j$-invariant over number fields of degree $p$, where $p$ is an odd prime. The question had been answered for $p=2$, so this paper completes the classification in case when the degree of the number field is prime.