{
  "id": 6381,
  "curve_key": "4316763434909420854962447601:-283695285706462634081551176762745225331401",
  "ainvs": [
    "1",
    "0",
    "0",
    "-89932571560612934478384325",
    "328351025123213146917128255360489330625"
  ],
  "rank_lower_bound": 11,
  "torsion": [
    3
  ],
  "family": null,
  "families": [],
  "naive_height": 190.89743870253207,
  "faltings_height": 13.635120509998133,
  "conductor": "322900376493751262239264260512012530",
  "bad_primes": [
    "2",
    "5",
    "7",
    "11",
    "13",
    "19",
    "37",
    "53",
    "67",
    "73",
    "313",
    "565537332343079892349"
  ],
  "discriminant": "-24605105574322783225316309560187844849129662903013869922644786361774592000000",
  "regulator": "306054717.468297473359606947892443340558080592752160119153013",
  "points": [
    [
      "5521675601400",
      "349800326261540175"
    ],
    [
      "9700171693170",
      "19201846749848997255"
    ],
    [
      "28294439765550",
      "142953088345341428175"
    ],
    [
      "77992483026850/9",
      "381632535311170301725/27"
    ],
    [
      "477454119850330/9",
      "-10276329807065671430515/27"
    ],
    [
      "5502846175330",
      "-315188045091754905"
    ],
    [
      "3330268004450",
      "8110880705989272375"
    ],
    [
      "126759523368100/9",
      "1163078642556085663975/27"
    ],
    [
      "12566964434290",
      "-34392660070653971065"
    ],
    [
      "659810187900150/121",
      "409857897884709139425/1331"
    ],
    [
      "21138830732022090/3481",
      "509869123378139070334725/205379"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/3 K3 family y^2+(7t^2+309)xy-91(t^2-1)(t^2-25)(t^2-121)y=x^3 (Layr-Labs 3-r12-z3k3), rational parameter search at t=1199/261. Integral model [1,0,0,a4,a6] as above. 11 points obtained by specializing polynomial sections at t; PARI/GP 2.19 ellrank bounds [11,11]; torsion Z/3Z (elltors); Neron-Tate regulator nonzero.",
  "created_at": "2026-11-01 21:49:16",
  "updated_at": "2026-11-01 21:49:16"
}