{
  "id": 5569,
  "curve_key": "0:-5893473044370290565085433242529819544",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "6821149356910058524404436623298402"
  ],
  "rank_lower_bound": 17,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell"
  ],
  "naive_height": 169.3338176453399,
  "faltings_height": 12.168807379569953,
  "conductor": "809086259038223489432482561001918463920931331342328985448022657409",
  "bad_primes": [
    "3",
    "7",
    "13",
    "71",
    "1129",
    "41103847444286828222618671"
  ],
  "discriminant": "-20100129933286586147971164262970660399187697064537478985485226878011787",
  "regulator": "5131924913845676431.26367610294808853810054460689456388229794496493466343",
  "points": [
    [
      "4498699234",
      "82590800958165817"
    ],
    [
      "70830989684",
      "84714287421362462"
    ],
    [
      "75576462962",
      "85163531694064505"
    ],
    [
      "-189029587838",
      "8167566285374905"
    ],
    [
      "-66811588298",
      "80764574961405289"
    ],
    [
      "-113190047348",
      "73286833485189985"
    ],
    [
      "-174520100830",
      "38803921122601926"
    ],
    [
      "-176563502608",
      "36288294931589409"
    ],
    [
      "149416266484",
      "100781449090811617"
    ],
    [
      "400016367230",
      "266137193847041801"
    ],
    [
      "698234096560",
      "589263790220224801"
    ],
    [
      "753797223634",
      "659648741022512377"
    ],
    [
      "899608279882",
      "857245404601634470"
    ],
    [
      "2296820583148/9",
      "4133912221319887675/27"
    ],
    [
      "29774245551856/9",
      "162480801815716227434/27"
    ],
    [
      "1971582005098888/2209",
      "87962112546412616493167/103823"
    ],
    [
      "2684278127044150/3481",
      "140103107061649046068667/205379"
    ]
  ],
  "submitter": "wgxli",
  "history": [],
  "commentary": "Specialization of a Mordell curve over Q(T) with generic rank 4.\n\nThe curve is a member of a universal parameterized rank-4 family\nthat includes every rank >= 4 Mordell curve over Q at least once up to isogeny.\nMoreover, this family achieves the minimum conductor degree of 12 for a\nnon-constant Mordell Q(T) curve of generic rank 4, \nand accounts for all such degree-12 curves up to Q(T)-isogeny.\n\nThe construction is as follows. Choose a rational point (a, b, c, d) on the Burkhardt quartic\n1 + a^3 + b^3 + c^3 + d^3 + 3abcd = 0\nsubject to appropriate non-singularity conditions.\n\nLet\nA := a^3 + 1,\nR := a^2 d^2 - bc,\nH(T) := bT^2 - cT - ad,\nG(T) := (b^3 - d^3)T^3 + 3bRT^2 - 3cRT - (2a^3 d^3 + c^3 + d^3).\nDefine the polynomial\nPhi_{a,b,c,d}(T) := A^2/4 (G(T)^2 + 4Ad^3 H(T)^3).\nThen the curve\nY^2 = X^3 + Phi_{a,b,c,d}(T)\nhas generic rank 4 over Q(T), and moreover every rank >= 4 Mordell curve over Q arises as some specialization.\n\nThis particular curve arises as (a, b, c, d, T) = (-43/36, -5/12, 1/4, 7/9, 4209911/23085591).\n",
  "created_at": "2026-10-23 11:48:15",
  "updated_at": "2026-10-23 11:48:15"
}