{
  "abstract": "In this paper, we give an explicit formula for the rank and generators (up to finite index) over $\\mathbb{Q}(t)$ of all non-trivial elliptic curves of the form $y^2=x^3+At^6+Bt^3+C$, which is a larger class of elliptic surfaces than the one in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024), namely $y^2=x^3+At^6+C$. Our proof provides a new method to find this formula using generators of the geometric Mordell-Weil group while this geometric Mordell-Weil group (which is a $\\mathbb{Z}[ω]$-module) no longer decomposes into rank-one submodules by the methods in Kloosterman's paper (2026) and Desjardins and Naskrecki's paper (2024). Moreover, our proof uses only a few elements in the Galois group of the coordinates of the generators and only light computations by hand.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.16349",
  "author_total": 1,
  "authors": [
    {
      "name": "Zhengheng Bao"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5c3vbmmb9aazq.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.16349",
  "fulltext_url": null,
  "id": "5c3vbmmb9aazq",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.16349",
  "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": "W7213427515",
  "pdf_url": "https://arxiv.org/pdf/2609.16349",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-30",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5c3vbmmb9aazq.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2609.16349",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.16349"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "A formula for the rank over $\\mathbb{Q}(t)$ of the elliptic curve $y^2=x^3+At^6+Bt^3+C$",
  "title_text": "A formula for the rank over ℚ(t) of the elliptic curve y^2=x^3+At^6+Bt^3+C",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/5c3vbmmb9aazq",
  "venue": "arXiv",
  "versions": []
}