{
  "id": 5511,
  "curve_key": "311856:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-6497",
    "0"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 37.95089046536527,
  "faltings_height": 1.2308146170885959,
  "conductor": "1350752288",
  "bad_primes": [
    "2",
    "73",
    "89"
  ],
  "discriminant": "17551675230272",
  "regulator": "124.6739037699364119233058370029856508147623832",
  "points": [
    [
      "-73",
      "292"
    ],
    [
      "-64",
      "392"
    ],
    [
      "-49",
      "448"
    ],
    [
      "-16",
      "316"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Quartic twist y^2 = x^3 - 6497x, obtained in my own bounded search in the family y^2 = x^3 + Ax with 0 < |A| <= 30000. Integer points were generated using A = (y^2 - x^3)/x for nonzero integer x between -173 and 5000, retaining integral A. This search produced A = -6497. Four independent rational points were recovered and checked exactly on this minimal model, establishing rank at least 4. The rational torsion subgroup is Z/2Z and the conductor is 1350752288.",
  "created_at": "2026-10-22 21:01:39",
  "updated_at": "2026-10-22 21:01:39"
}