{
  "id": 4576,
  "curve_key": "418786222199487531343516686574027473961986081:-8567460198167502655502891464219069024449724164991161756801232311729",
  "ainvs": [
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  ],
  "rank_lower_bound": 27,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 308.23780345776936,
  "faltings_height": 23.419301740112044,
  "conductor": "34984200010319809978334193632379682476220287211750140365628784855559193613974153895640025232509783596107677770490",
  "bad_primes": [
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    "5",
    "17",
    "29",
    "103",
    "5431",
    "836654506425293460683756935665429791599",
    "1684687433819774755654444078936826242963527477936183934540244411"
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  "discriminant": "26706629531322531858945378632699327672220905127683241133082203465330267181688709593638973901891276758922495402071134763456000000",
  "regulator": "3990140741840927527195879309970.4477953840369748481843994091125341455689079284073641626492999",
  "points": [
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      "-36991618186498779582799124096476"
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    [
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      "-2129467887225005475946739095846"
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      "1781576823688095194071",
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      "-44695435257456863202207698202676"
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    [
      "14865575375418639479359/9",
      "91326385280381013781239003903148/27"
    ],
    [
      "1755873931072363438351",
      "-3173158458442715675987048682076"
    ],
    [
      "197693018337399474063271/121",
      "-6340524472427208371469121099492556/1331"
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  ],
  "submitter": "apkorte",
  "history": [
    {
      "kind": "primes_recorded",
      "user": "Daniel Błażewicz",
      "at": "2026-10-18 06:48:00"
    }
  ],
  "commentary": "Specialization at T = -660/137 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -485042948797679528112T^8 + 12463578805199058894528T^7 - 230629938307243679610144T^6 - 1586012147509225586376096T^5 - 3868907398105357015696080T^4 - 6794745991541079206248224T^3 - 4791118686138149116505904T^2 + 6061871481875161826582772T - 1374235222299991775805627,\nA6 = 3820001086939854414872700973440T^12 - 155249980524531925200471635032320T^11 + 5279785372691676756614905714258560T^10 + 22439305421110286908829252745667200T^9 + 198311921848923368813090766393603840T^8 + 1955108933617356512048828761030033920T^7 + 6903605561322701279549833535328289920T^6 + 8139034537986456490540355512081445520T^5 - 1317692978359257302156310289643874060T^4 - 2476393922218195353910049397638432280T^3 + 5841882186136964852470754649206854356T^2 - 3899452237007593054578752344925650716T + 735364979113660346489805816953176854.\nIt is a 2-neighbour of the fibration given in curve#3900: its Mordell-Weil lattice (rank 17, no reducible fibres) is a root-free Kneser 2-neighbour of that fibration's, realised by the pencil |O + P - F| for a section P of height 12. Constructed and searched by Claude (Anthropic): a Mestre-Nagao sieve over the fibration, then a 2-covering point search seeded with its generic sections.",
  "created_at": "2026-10-18 06:30:45",
  "updated_at": "2026-10-18 06:48:00"
}