Explicit isogenies of prime degree over number fields

TL;DR

This paper presents an explicit algorithm for classifying prime degree isogenies of elliptic curves over number fields, implementing it in Sage and PARI/GP, and discovering new instances over cubic and quadratic fields.

Contribution

It provides the first explicit, implementable algorithm for prime degree isogenies over number fields, including new classifications for cubic and quadratic fields.

Findings

First instances of prime degree isogenies over cubic fields identified.

Algorithm implemented in Sage and PARI/GP for practical use.

Correctness proven unconditionally for semistable elliptic curves.

Abstract

We provide an explicit and algorithmic version of a theorem of Momose classifying isogenies of prime degree of elliptic curves over number fields, which we implement in Sage and PARI/GP. Combining this algorithm with recent work of Box-Gajovi\'c-Goodman we determine the first instances of isogenies of prime degree for cubic number fields, as well as for several quadratic fields not previously known. While the correctness of the general algorithm relies on the Generalised Riemann Hypothesis, the algorithm is unconditional for the restricted class of semistable elliptic curves.