{
  "id": 4213,
  "curve_key": "3337526316063796944070790612584955559376:-380446680337192823105118964434239138649906168994489393551040",
  "ainvs": [
    "0",
    "1",
    "0",
    "-69531798251329103001474804428853240820",
    "440331805945825026718858572381778190955418316341411643744"
  ],
  "rank_lower_bound": 25,
  "torsion": [],
  "naive_height": 274.3773926752466,
  "faltings_height": 20.909591692787536,
  "conductor": "28528256290887068868408626166511695602057583892992150721316098304212624288718334294312692700962664",
  "bad_primes": [
    "2",
    "3",
    "11",
    "23",
    "59",
    "73",
    "149",
    "117404958347",
    "360052418004907",
    "116546999415497796840397",
    "1486035063747424426354902812453456170193"
  ],
  "discriminant": "-62246931462256159527948079123946250057670054353474258707783128071266665842110156793590299032164096451695314138967808",
  "regulator": "2709767040418178959994292806.847007372676059204824957589297661901879834577009768833013055",
  "points": [
    [
      "-1459744115271345976",
      "-23210341214905210122871071408"
    ],
    [
      "5653961798739909935",
      "15097797107617150961062779762"
    ],
    [
      "8866158697127310782",
      "22821254968493534662184042286"
    ],
    [
      "-4159341641238658225",
      "-25643345886875315503909459338"
    ],
    [
      "4715383166427561314",
      "-14741387915289283297627360002"
    ],
    [
      "256594228528237698488",
      "-4108152425674516641672982875000"
    ],
    [
      "6558398127820761788",
      "16322017460630314703243075520"
    ],
    [
      "4194360568451337212",
      "14915774700973485736476067824"
    ],
    [
      "425546690950326405146",
      "-8776849986627394579012267368426"
    ],
    [
      "13363583060209697474",
      "-43562346496087012106387604642"
    ],
    [
      "15467455623189646796",
      "55365329023526088513086458824"
    ],
    [
      "5311266752748500582",
      "-14861303666251731957583288314"
    ],
    [
      "21695765071238691119",
      "-95624880950299229850858779022"
    ],
    [
      "-861329766939415365886/121",
      "-31904706171537968406786173280042/1331"
    ],
    [
      "3175204864739458622",
      "-15860841155270850357708933294"
    ],
    [
      "36595776963029132886566/361",
      "6978540806724545505200747993376366/6859"
    ],
    [
      "4951500174843122246",
      "-14745941044001192380429258074"
    ],
    [
      "7043645820058257470",
      "17321387607045149710505728338"
    ],
    [
      "-7188355424665278562",
      "23847664920302019301086818610"
    ],
    [
      "-10473263363264475208",
      "4444354302070502813671476216"
    ],
    [
      "-17640343433865216865426/1681",
      "261244345321199240016155222671698/68921"
    ],
    [
      "543048233879840572178/169",
      "-34743416163451673102758210133322/2197"
    ],
    [
      "10605731547048823322",
      "29930674139187979110654812694"
    ],
    [
      "326575073057219973646940/38809",
      "162380378473897364382743701319025024/7645373"
    ],
    [
      "1045911968447975227019/49",
      "-31958572350425396593214730255768/343"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -120/121 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -18808819639564736183472T^8 - 270390604164314807890368T^7 - 1064698058908765727554464T^6 - 690609682722857600525472T^5 - 1141985348588431905168240T^4 - 1276711546748486547364128T^3 + 1533178281333365507000016T^2 + 1003847548076768965363668T - 662667526757651887035567,\nA6 = 198461588427944563947255082869120T^12 + 5470876228138813326458542895357184T^11 + 78310570892779922090226615255906432T^10 + 585449236963957414662829259015483520T^9 + 1854487489939448014773845858584024704T^8 + 1020492248925323261478020126191388928T^7 - 2397613146941260720543566338947934208T^6 - 1992669351542649345334637022139959408T^5 + 1587582544452083911151305726515006564T^4 + 1461616308592226634206845276455690360T^3 - 684661031717307107294832767505632268T^2 - 464296295429065890603785053139616684T + 208515476578669742594974588404489810.\nIt is a 2-neighbour of the fibration given in curve#4191 (itself a 2-neighbour of the fibration 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": "2026-10-14 02:05:05",
  "updated_at": "2026-10-14 02:05:05"
}