{
  "id": 6367,
  "curve_key": "616448786671639936:-10129283865424450725899865184",
  "ainvs": [
    "0",
    "1",
    "0",
    "-12842683055659165",
    "11723708173293701395682900"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 128.97045626434178,
  "faltings_height": 8.805353033512215,
  "conductor": "1853422857999600420",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "19",
    "29",
    "71",
    "79",
    "89",
    "199",
    "43411"
  ],
  "discriminant": "-59240819139853690897439027053658537224100031904782000",
  "regulator": "285228.28352943303094018254008726425057515624475305",
  "points": [
    [
      "-6724234230655/27889",
      "4176135225819273405/4657463"
    ],
    [
      "-1899340040615/74529",
      "70583146428309437285/20346417"
    ],
    [
      "-384676329187/2116",
      "276171946115032059/97336"
    ],
    [
      "-975752135/9",
      "92911842894455/27"
    ],
    [
      "27334670113/441",
      "30945893980272209/9261"
    ],
    [
      "1505304139075865/201601",
      "58397260513168328661195/90518849"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/6-torsion Kubert–Tate family curve from my rational-parameter search of y^2+(1-c)xy-by=x^3-bx^2, with b=c(c+1), following the construction of David Renshaw on curve#1192 and the c=1/6916 example of Lewis Combes on curve#5970.\r\n\r\nHere c=-7175/119209. An integral model is [126384,803843950,95825433435550,0,0], which reduces to the displayed global minimal model.\r\n\r\nThe completed rank-6 batch identified this specialization. A separate rank computation on the minimal model returned bounds [6,6] and the six displayed rational points. All six points were checked exactly on this model. The minimal discriminant was fully factored and every listed bad-reduction prime proved prime.",
  "created_at": "2026-11-01 14:35:40",
  "updated_at": "2026-11-01 14:35:40"
}