Classification of torsion of elliptic curves over sextic fields

Abstract

We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $6$. Apart from the $44$ groups that occur infinitely often, which were determined by Derickx and Sutherland, exactly four groups occur: ℤ/25ℤ, ℤ/37ℤ, ℤ/4ℤ×ℤ/12ℤ and ℤ/2ℤ×ℤ/24ℤ. The first three were known to occur, while the group ℤ/2ℤ×ℤ/24ℤ is new. We also show that, up to isomorphism, each of the groups ℤ/37ℤ and ℤ/2ℤ×ℤ/24ℤ is the torsion group of a single elliptic curve over a single sextic field. The methods build on those developed by Derickx and Najman for quartic fields and by the author for quintic fields. Among the new ingredients are a formal immersion criterion at the prime $2$ which uses the Cartier operator, and arguments with several diamond operators and with cuspidal divisor classes on quotients of modular curves.

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BibTeX RIS

BibTeX

@misc{najman2026classification,
  title = {{Classification of torsion of elliptic curves over sextic fields}},
  author = {Filip Najman},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.03933},
  eprint = {2611.03933},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.03933},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Classification of torsion of elliptic curves over sextic fields
AU  - Filip Najman
PY  - 2026
DA  - 2026/11/02
PB  - arXiv
DO  - 10.48550/arxiv.2611.03933
UR  - https://arxiv.org/abs/2611.03933
AB  - We determine all the groups that appear as the torsion group of an elliptic curve over a number field of degree $6$. Apart from the $44$ groups that occur infinitely often, which were determined by Derickx and Sutherland, exactly four groups occur: ℤ/25ℤ, ℤ/37ℤ, ℤ/4ℤ×ℤ/12ℤ and ℤ/2ℤ×ℤ/24ℤ. The first three were known to occur, while the group ℤ/2ℤ×ℤ/24ℤ is new. We also show that, up to isomorphism, each of the groups ℤ/37ℤ and ℤ/2ℤ×ℤ/24ℤ is the torsion group of a single elliptic curve over a single sextic field. The methods build on those developed by Derickx and Najman for quartic fields and by the author for quintic fields. Among the new ingredients are a formal immersion criterion at the prime $2$ which uses the Cartier operator, and arguments with several diamond operators and with cuspidal divisor classes on quotients of modular curves.
ER  -

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