{
  "abstract": "For any primes $p \\neq \\ell$, $\\ell$-isogeny graphs of ordinary elliptic curves defined over $\\mathbb{F}_{p^2}$ have a typical structure called $\\ell$-volcanoes, and the structure is the core of Sutherland's supersingularity testing algorithm for elliptic curves. In this paper, by exploiting the properties of $3$-isogenies between twisted Hessian curves, we show that when $p \\equiv 2 \\pmod{3}$ and $\\ell = 3$, every ordinary twisted Hessian curve defined over $\\mathbb{F}_p$ lies on the surface of the $3$-volcano. As an application, we give an improved version of Sutherland's supersingularity testing algorithm specialized to twisted Hessian curves defined over $\\mathbb{F}_p$ with $p \\equiv 2 \\pmod{3}$. We also give a generalization of the known fact that any supersingular $j$-invariant is a cube in $\\mathbb{F}_{p^2}$; we show that for any twisted Hessian curve $H(a,d)$ defined over $\\mathbb{F}_{p^2}$, its $j$-invariant is not a cube in $\\mathbb{F}_{p^2}$ if and only if $H(a,d)$ is ordinary and lies on the floor of a $3$-volcano.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.11037",
  "author_total": 2,
  "authors": [
    {
      "name": "Yuji Hashimoto"
    },
    {
      "name": "Koji Nuida"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eq0f7xmd75ehr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.11037",
  "fulltext_url": null,
  "id": "eq0f7xmd75ehr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.11037",
  "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.NT"
    ],
    [
      "openalex",
      "1710|Cryptography and Residue Arithmetic"
    ]
  ],
  "openalex": "W7212247834",
  "pdf_url": "https://arxiv.org/pdf/2609.11037",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-10",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eq0f7xmd75ehr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:30Z",
      "native_id": "2609.11037",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.11037"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields",
  "title_text": "Ordinary 3-Isogeny Graphs and Improvement of Supersingularity Testing for Twisted Hessian Curves over Prime Fields",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:30Z",
  "url": "https://research.draconitic.com/paper/eq0f7xmd75ehr",
  "venue": "arXiv",
  "versions": []
}