{
  "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: $\\mathbb{Z}/28\\mathbb{Z}$, $\\mathbb{Z}/30\\mathbb{Z}$ and $\\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/18\\mathbb{Z}$. 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 $\\mathbb{Z}/28\\mathbb{Z}$ and $\\mathbb{Z}/30\\mathbb{Z}$ were found by van Hoeij, while the group $\\mathbb{Z}/2\\mathbb{Z}\\times\\mathbb{Z}/18\\mathbb{Z}$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12846",
  "author_total": 1,
  "authors": [
    {
      "name": "Filip Najman"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/21jvh0zq6w4q6.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12846",
  "fulltext_url": null,
  "id": "21jvh0zq6w4q6",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12846",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AG"
    ],
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7212565343",
  "pdf_url": "https://arxiv.org/pdf/2609.12846",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/21jvh0zq6w4q6.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:50Z",
      "native_id": "2609.12846",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12846"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.AG",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Classification of torsion of elliptic curves over quintic fields",
  "title_text": "Classification of torsion of elliptic curves over quintic fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:50Z",
  "url": "https://research.draconitic.com/paper/21jvh0zq6w4q6",
  "venue": "arXiv",
  "versions": []
}