{
  "id": 4196,
  "curve_key": "78537807195709988908808850559060270116428961:-999467947922826921100839536103208935198718130306522691250009213041",
  "ainvs": [
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    "1",
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    "1156791606392160788311565921568969479338897882099581553779552249"
  ],
  "rank_lower_bound": 27,
  "torsion": [],
  "naive_height": 303.9401678878798,
  "faltings_height": 23.353739973129777,
  "conductor": "2354916239112007892073152869234207119051588829446921311745649592265246092225042640809243177031557188527240414952810",
  "bad_primes": [
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    "5",
    "7",
    "11",
    "17",
    "6089",
    "45589",
    "88431538658940336339975361639116868405067",
    "666238784557858910525818998986943572963543744552344842686417"
  ],
  "discriminant": "-297743220529971557422610830343358680816070984996754776293483102932642726894292413999595938037364363301326873960258312657152000000",
  "regulator": "6931090223496514928578101729726.3202399948415484023659143849970968015863772112105095849527082",
  "points": [
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      "-23603202030014569915902684227704"
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    [
      "1130820936257276474477",
      "27433219914898873424338642197026"
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      "-23978033884145110898784281245908"
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      "-42425012383953089098408291684804"
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      "4226408575616889480327",
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      "-388018098621624591070028878938804"
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    [
      "4687396896923564933607",
      "-310607924154523149115686674542804"
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    [
      "122184542682401365902583/169",
      "41204730135484100920286453294031612/2197"
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      "11955852221994028703833/9",
      "983747803908506954614902566084432/27"
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    [
      "-2088897983340399110220337/1369",
      "-508667413342621650603351597557716612/50653"
    ],
    [
      "19569270897037291414637/16",
      "-2008842429432523926702663257730781/64"
    ],
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      "870165267558112173607",
      "-19796509385034880897769140902804"
    ],
    [
      "47827993433848136492297967/3481",
      "329405342026900493186686330527718013284/205379"
    ],
    [
      "2625516714328936065903/4",
      "152967736762800852061665463574943/8"
    ],
    [
      "3120117263748949907151963/2809",
      "3966385721092977515658355892654188192/148877"
    ],
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      "3540791058480848407764127/361",
      "-6609918563989598910339647893428728636/6859"
    ],
    [
      "-107683085987849143813",
      "-36492937418767493576570942880884"
    ],
    [
      "10894375924188849344165845561543/18661472449",
      "51021048931240204527627884088089920007355469428/2549287766840543"
    ],
    [
      "-100619538762485959412193/121",
      "58660323476416373544527204685242676/1331"
    ],
    [
      "611566000835364676833103/729",
      "-380937767694049945481313064729043132/19683"
    ],
    [
      "873283662534589760887",
      "19847128211153255150718993005916"
    ]
  ],
  "submitter": "apkorte",
  "history": [
    {
      "kind": "primes_recorded",
      "user": "Daniel Błażewicz",
      "at": "2026-10-12 05:04:39"
    }
  ],
  "commentary": "Specialization at T = -257/534 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -4247232044073407555787T^8 + 31562440352380138911048T^7 - 1079699554459262036768544T^6 + 808673010384011405337504T^5 - 370492253920133220842280T^4 - 583074314969616668304144T^3 + 1749298418924942437857936T^2 + 6981774061823935535232T - 733152591313571732712192,\nA6 = 99353320243156880097765275026566T^12 - 1192047168420920079142624551372696T^11 + 51816962014333011197785218484470384T^10 - 160733095313938224161654547702105120T^9 + 1161258380951665950605765814917343480T^8 - 500375410396179478662655437513305328T^7 - 2173656091315178125605895871772822192T^6 + 963181788398399075637606795285979968T^5 + 1772421005485324185894755005716179280T^4 - 460279153080260982936199267860712320T^3 - 927109903404780261014801473928185344T^2 + 58145438007481575316035550439430144T + 241998303936746269490887631501697024.\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": "2026-10-12 04:52:15",
  "updated_at": "2026-10-12 05:04:39"
}