Explicit isogenies of prime degree over number fields

Banwait, Barinder S.; Derickx, Maarten

doi:10.2140/ant.2025.19.1147

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Explicit isogenies of prime degree over number fields

Barinder S. Banwait and Maarten Derickx

Vol. 19 (2025), No. 6, 1147–1197

DOI: 10.2140/ant.2025.19.1147

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, Gajović and Goodman we obtain the first classifications of the possible prime degree isogenies of elliptic curves over 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.

Keywords

elliptic curves, isogenies, prime degree

Mathematical Subject Classification

Primary: 11G05

Secondary: 11G15, 11Y60

Milestones

Received: 19 October 2022

Revised: 28 March 2024

Accepted: 3 September 2024

Published: 14 May 2025

Authors

Barinder S. Banwait
Department of Mathematics and Statistics Boston University Boston, MA United States

Maarten Derickx
Department of Mathematics University of Zagreb Zagreb Croatia

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Vol. 19, No. 6, 2025