{
  "id": 5659,
  "curve_key": "0:659716488",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "-763561"
  ],
  "rank_lower_bound": 3,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell",
    "cubesum"
  ],
  "naive_height": 40.61464147411898,
  "faltings_height": 1.4422093653015442,
  "conductor": "247393683",
  "bad_primes": [
    "3",
    "1009"
  ],
  "discriminant": "-251866808182323",
  "regulator": "64.798821256331517011074516797479204411426060",
  "points": [
    [
      "1009/4",
      "31275/8"
    ],
    [
      "1009/9",
      "21680/27"
    ],
    [
      "1744561/1936",
      "2303002935/85184"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Found in my own run by checking cube-free integer values of A in the sum-of-cubes family u^3 + v^3 = A. The specialization A = 9081 gives the global minimal model Y^2 + Y = X^3 - 763561.\r\n\r\nThe explicit map is u = (A + 6Y + 3)/(6X) and v = (A - 6Y - 3)/(6X), with A = 9081. The three submitted points correspond to (u,v) = (43/2,-19/2), (62/3,19/3), and (2410279/76076,-2154727/76076).\r\n\r\nAll three points were checked exactly and are independent. Matching rank bounds [3,3] establish rank exactly 3, with trivial rational torsion and conductor 247393683.",
  "created_at": "2026-10-24 00:53:52",
  "updated_at": "2026-10-24 00:53:52"
}