Classification of torsion of elliptic curves over quintic fields
Abstract
We determine all the groups that appear as the torsion group of an elliptic curve over a quintic number field. Apart from the groups that already occur infinitely often, which were determined by Derickx and Sutherland, exactly three groups occur: ℤ/28ℤ, ℤ/30ℤ and ℤ/2ℤ×ℤ/18ℤ. Up to isomorphism of the pair (K,E), the first and the third are each realised by a single elliptic curve and the second by two curves, which are $2$-isogenous over a common quintic field. The curves realising ℤ/28ℤ and ℤ/30ℤ were found by van Hoeij, while the group ℤ/2ℤ×ℤ/18ℤ is new, and $5$ is the smallest degree in which a non-cyclic sporadic torsion group occurs. The methods improve on those developed by Derickx and Najman, and are based on Hecke sieves and the arithmetic of cuspidal divisor classes.
Related
- Classification of torsion of elliptic curves over sextic fields
- Torsion groups of rational elliptic curves over ℤp-extensions of quadratic fields: the p≤ 5 case
- Torsion growth of rational elliptic curves over Zp-extensions of quadratic fields
- Tamagawa numbers and torsion of elliptic curves over function fields
- Fields of definition of p-torsion points of elliptic curves and their ramification
BibTeX
@misc{najman2026classification,
title = {{Classification of torsion of elliptic curves over quintic fields}},
author = {Filip Najman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.12846},
eprint = {2609.12846},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12846},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Classification of torsion of elliptic curves over quintic fields
AU - Filip Najman
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2609.12846
UR - https://arxiv.org/abs/2609.12846
AB - We determine all the groups that appear as the torsion group of an elliptic curve over a quintic number field. Apart from the groups that already occur infinitely often, which were determined by Derickx and Sutherland, exactly three groups occur: ℤ/28ℤ, ℤ/30ℤ and ℤ/2ℤ×ℤ/18ℤ. Up to isomorphism of the pair (K,E), the first and the third are each realised by a single elliptic curve and the second by two curves, which are $2$-isogenous over a common quintic field. The curves realising ℤ/28ℤ and ℤ/30ℤ were found by van Hoeij, while the group ℤ/2ℤ×ℤ/18ℤ is new, and $5$ is the smallest degree in which a non-cyclic sporadic torsion group occurs. The methods improve on those developed by Derickx and Najman, and are based on Hecke sieves and the arithmetic of cuspidal divisor classes.
ER -