{
  "abstract": "We study the elliptic family $E_t:Y^2=X^3+4X^2+16t^2$ arising from rational $2$-cycles of $x^3+bx+a$. For every $t\\in\\mathbb{Q}^\\times$, we prove that $P_t=(0,-4t)$ has infinite order, so every nonsingular rational fiber has positive rank. We determine its rational torsion subgroup: it is cyclic of order $2$ precisely when $t=u(u^2+4)/4$ for some $u\\in\\mathbb{Q}^\\times$, and is trivial otherwise. No such fiber admits a rational $3$- or $5$-isogeny. An explicit birational dictionary then shows that, for every fixed $a\\in\\mathbb{Q}^\\times$, infinitely many $b\\in\\mathbb{Q}$ yield a rational $2$-cycle of $x^3+bx+a$. The uniform non-torsion assertion follows from Nagell--Lutz integrality. The torsion exclusions combine elementary $2$-descent with explicit genus-$3$ curves, an unconditional rank-zero Prym argument, and two-cover descent with elliptic Chabauty. Exact Magma and SageMath certificates accompany the computer-assisted steps.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2606.15109",
  "author_total": 3,
  "authors": [
    {
      "name": "Sompong Chuysurichay"
    },
    {
      "name": "Sawian Jaidee"
    },
    {
      "name": "Chatchawan Panraksa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/3f7rzmnfwdesc.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.15109",
  "fulltext_url": null,
  "id": "3f7rzmnfwdesc",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.15109",
  "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",
      "2608|Algebraic Geometry and Number Theory"
    ]
  ],
  "openalex": "W7164900997",
  "pdf_url": "https://arxiv.org/pdf/2606.15109",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-05",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/3f7rzmnfwdesc.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:57Z",
      "native_id": "2606.15109",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.15109"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles",
  "title_text": "Torsion and Positive Rank in an Elliptic Family Arising from Cubic 2-Cycles",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:57Z",
  "url": "https://research.draconitic.com/paper/3f7rzmnfwdesc",
  "venue": "arXiv",
  "versions": []
}