{
  "abstract": "For an integer $t\\geq-1$, let $θ_t$ be the largest real root of $g_t(X)=X^3-tX^2-(t+3)X-1$, and set $R_t=\\mathbb{Z}[θ_t]\\subseteq\\mathcal{O}_t=\\mathcal{O}_{\\mathbb{Q}(θ_t)}$, $N_t=[\\mathcal{O}_t:R_t]$, and $\\varepsilon_t=[\\mathcal{O}_t^\\times:R_t^\\times]$. We determine $\\varepsilon_t$ when $N_t$ is squarefree, $27$, or $343$: the only nontrivial indices in these cases are $\\varepsilon_3=3$, $\\varepsilon_5=7$, $\\varepsilon_{12}=13$, and $\\varepsilon_{54}=19$. Local conductor calculations give $\\varepsilon_t\\mid N_t$ for squarefree $N_t$. For arbitrary $N_t$, a regulator comparison shows that $t\\geq2N_t$ implies $\\varepsilon_t=1$. When $N_t=p^3$ with $p\\neq3$ a rational prime, this bound and the index criterion leave at most four possible parameters with $\\varepsilon_t>1$ for each fixed $p$. For arbitrary additive index, we also prove that $13\\mid\\varepsilon_t$ if and only if $t=12$ or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When $N_t$ is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over $\\operatorname{Pic}(\\mathcal{O}_t)$, and the corresponding integral matrix-conjugacy classes.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.30637",
  "author_total": 1,
  "authors": [
    {
      "name": "Junyu Lu"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/15bajage2caq5.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.30637",
  "fulltext_url": null,
  "id": "15bajage2caq5",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.30637",
  "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": "W7204933426",
  "pdf_url": "https://arxiv.org/pdf/2608.30637",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/15bajage2caq5.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:19:58Z",
      "native_id": "2608.30637",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.30637"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Unit Indices of Shanks Orders",
  "title_text": "Unit Indices of Shanks Orders",
  "type": "preprint",
  "updated_at": "2026-10-30T12:19:58Z",
  "url": "https://research.draconitic.com/paper/15bajage2caq5",
  "venue": "arXiv",
  "versions": []
}