{
  "id": 4206,
  "curve_key": "271967257668904150894637268279305080801:-4509806805700491661182025182252401860043200697432274420049",
  "ainvs": [
    "1",
    "0",
    "0",
    "-5665984534768836476971609755818855850",
    "5219683802894087570340363768228062076043107401635783332"
  ],
  "rank_lower_bound": 23,
  "torsion": [],
  "naive_height": 265.50720923276833,
  "faltings_height": 19.965087442924965,
  "conductor": "723978752176623467066621027541941900700214159357362242896740061914841236010230",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "59",
    "103",
    "13993487",
    "183442122610296247039350106173500616764948750117248806629235397"
  ],
  "discriminant": "-128458203615624650995216763709151639278354750581132539799094319899995823280281675468797466643613214082560000000",
  "regulator": "10455661477994131633140.2451443233221112054969299275762773977703428414390824336950156",
  "points": [
    [
      "535037009788386324",
      "-1530141999064567770617117082"
    ],
    [
      "871499077260724404",
      "-971440339546385672157878202"
    ],
    [
      "1292929246504717404",
      "235186301219037676665564798"
    ],
    [
      "1362038657239825524",
      "170818329612928103707562118"
    ],
    [
      "6410478749958461124",
      "15242432497262818646246761158"
    ],
    [
      "1593871716832107204",
      "487797219508385264889582198"
    ],
    [
      "1546286337888240804",
      "-394489778447416176370147002"
    ],
    [
      "1725132876083074404",
      "761082774624188303305971798"
    ],
    [
      "-359727259528087596",
      "2685394357307401282322277798"
    ],
    [
      "1564457609368898904",
      "429586287266292359412347298"
    ],
    [
      "1991968440935797452",
      "-1355444294058806350373658930"
    ],
    [
      "1797953579871143004",
      "-919041908104639108713097602"
    ],
    [
      "1406068099505663904",
      "180994464648238591876706298"
    ],
    [
      "1973566236105780444",
      "-1313185767303671429344092162"
    ],
    [
      "365243040499058304",
      "-1788559954699243754643007002"
    ],
    [
      "2584999796402003724",
      "-2801189476789864920022269042"
    ],
    [
      "1297357260502852404",
      "-229142225016502183040702202"
    ],
    [
      "1941630515534440404",
      "-1240259905874907195231182202"
    ],
    [
      "1808332337875175604",
      "941842106791244424822427398"
    ],
    [
      "1346625314710326324",
      "178032549135765467262850758"
    ],
    [
      "1423586776796530404",
      "196731091817191196559561798"
    ],
    [
      "-520942407659263098",
      "-2833718680339335974348124480"
    ],
    [
      "1345399794628820604",
      "178824938534227808941816998"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -186/163 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -32314259208676675198192T^8 - 220263400022708733054624T^7 - 255006976139783552896608T^6 + 705363094453509328737024T^5 + 1047835198424784109021200T^4 + 1820453244371835449110416T^3 - 1245666920662047666014424T^2 - 3926333297851033484227032T - 1657966129723797525563067,\nA6 = 2220158269397315819350729045585024T^12 + 23424072793928593017661886421060992T^11 + 69989708296427149663267141905289920T^10 - 61968539237944624284964605273911360T^9 - 599227600313137750772024638621900656T^8 - 203890468677123352982190860910686112T^7 + 2258581157822950043811972713225053920T^6 + 559769644036858048987416888208533696T^5 - 3908030961820075418207725091990624736T^4 - 1985850721280648599946097200370542160T^3 + 2921751862616849299905524795576315928T^2 + 3089933922534003123866046309582827784T + 831153037936820603440551361962407190.\nIt is a 2-neighbour of the fibration given in curve#4187 (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": "2026-10-13 21:11:59",
  "updated_at": "2026-10-13 21:11:59"
}