{
  "abstract": "Let $E_{A,B}\\colon y^2 +Axy + By = x^3$ be an elliptic curve over $\\mathbb{Q}$. The $3$-torsion point $(0,0)$ induces a $3$-isogeny $ϕ\\colon E_{A,B}\\to E_{A,B}'$. Assuming $E_{A,B}$ has good reduction at $3$, we construct an explicit $(m+t)\\times m$ matrix $M_{A,B}'$ over $\\mathbb{F}_3$, whose kernel encodes the dual isogeny Selmer group $\\operatorname{Sel}_{\\widehatϕ}(E_{A,B}')$ modulo the image of the torsion point $(0,0)$. Here, $m = ω(B) - 1$, and $t$ encodes the \\emph{global Selmer ratio} $3^{t-2}$. We compute $M_{A,B}'$ in various regimes for billions of elliptic curves $E_{A,B}$. Based on our data, and motivated by prevalence of random linear algebraic models throughout number theory, we conjecture that, for fixed $m$ and $t$, the matrices $M_{A,B}'$ become uniformly distributed, and we formulate a corresponding conjecture for the distribution of $\\operatorname{Sel}_ϕ(E_{A,B})$. Our model predicts that, for fixed $t$, the average size of $\\operatorname{Sel}_ϕ(E_{A,B})$ is $1 + 3^t$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.31330",
  "author_total": 2,
  "authors": [
    {
      "name": "Ariel Weiss"
    },
    {
      "name": "Dongchen Zou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1j449h7sykj8d.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.31330",
  "fulltext_url": null,
  "id": "1j449h7sykj8d",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.31330",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.NT"
    ],
    [
      "openalex",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7214582696",
  "pdf_url": "https://arxiv.org/pdf/2609.31330",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-25",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1j449h7sykj8d.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:24:57Z",
      "native_id": "2609.31330",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.31330"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point",
  "title_text": "Experiments on $3$-isogeny Selmer groups of elliptic curves with a $3$-torsion point",
  "type": "preprint",
  "updated_at": "2026-10-30T12:24:57Z",
  "url": "https://research.draconitic.com/paper/1j449h7sykj8d",
  "venue": "arXiv",
  "versions": []
}