{
  "abstract": "Let $G=C_7\\times C_7$, and let $S(G)$ denote the set of integer values of its group determinant. We determine $S(G)$ by classifying the values divisible by $7$; the coprime values are already known. Every nonzero divisible value has valuation at least $10$, and every multiple of $7^{12}$ occurs. At valuations $10$ and $11$, we give necessary and sufficient conditions on the cofactor in terms of two additive invariants of ideals in $\\mathbb{Z}[ζ_7]$. The first condition is a bounded signed sum of prime-ideal invariants. The second requires a prime ideal with nonzero invariant pair whose norm divides the cofactor. Neither cofactor set is a union of congruence classes modulo any positive integer. The least positive divisible value is $43\\cdot7^{10}$, and the least positive value of valuation $11$ is $8\\cdot7^{11}$. The proof combines integral reconstruction from character values with a calculation of the global-unit image modulo $7$. We conclude by identifying the additional local conditions and realization problems that arise at primes at least $11$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.09542",
  "author_total": 1,
  "authors": [
    {
      "name": "Chatchawan Panraksa"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1aytt493cwj04.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.09542",
  "fulltext_url": null,
  "id": "1aytt493cwj04",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.09542",
  "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",
      "2607|Finite Group Theory Research"
    ]
  ],
  "openalex": "W7212156141",
  "pdf_url": "https://arxiv.org/pdf/2609.09542",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-08",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1aytt493cwj04.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:20:15Z",
      "native_id": "2609.09542",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.09542"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.NT",
      "subfield": "algebra-number-theory"
    }
  ],
  "title": "Integer group determinants for the elementary abelian group of order 49",
  "title_text": "Integer group determinants for the elementary abelian group of order 49",
  "type": "preprint",
  "updated_at": "2026-10-30T12:20:15Z",
  "url": "https://research.draconitic.com/paper/1aytt493cwj04",
  "venue": "arXiv",
  "versions": []
}