{
  "id": 4215,
  "curve_key": "64563668768127909134174051937225588977761:-14289186708674894489872146531557693356377120413768518080372241",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1345076432669331440295292748692199770370",
    "16538410542447794548351042856950811912819281638729953352900"
  ],
  "rank_lower_bound": 25,
  "torsion": [],
  "naive_height": 281.9054114291349,
  "faltings_height": 21.478747555586423,
  "conductor": "145159370006636113651476543950798325070663960016825995909894028793437575072639958539379467830",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "31",
    "41",
    "191",
    "9463",
    "272045818228399",
    "1689549936504702447779",
    "269289845010499341380794935162012655492511"
  ],
  "discriminant": "37587203028827978934621409770806440165557509777604684996745791608004856566183788027112976340378386506108095053824000000",
  "regulator": "120199129353076184786706481.4146464666783363046708490151304595531311902876665884507251156",
  "points": [
    [
      "31022102829197834140",
      "-68308703971778626160786294570"
    ],
    [
      "27254107099871058340",
      "-11114505313443161507032290770"
    ],
    [
      "130605568740969728140",
      "-1438300912098667340614665620570"
    ],
    [
      "1082941743889928335276",
      "35617323216777934161209878161742"
    ],
    [
      "58021259030717860",
      "-128297964557766239464892315090"
    ],
    [
      "747682451106726169795/4",
      "-20073455761726752985814383181605/8"
    ],
    [
      "32211792971574143140",
      "81449559052721187190686384430"
    ],
    [
      "-7704446875028354588",
      "-162616592133163642817927280338"
    ],
    [
      "-10139947296975550940",
      "-170689308953475762628103970770"
    ],
    [
      "-13637045145696794172710/361",
      "795271869903149034610410942458620/6859"
    ],
    [
      "3101418332911007140",
      "-111340009630837989618810863570"
    ],
    [
      "45396421470982098940",
      "-221430106096393160118811408370"
    ],
    [
      "553221038200018114435/4",
      "12588395825160296840827073268815/8"
    ],
    [
      "5159851082359501711060",
      "-370633616156087213518472553000770"
    ],
    [
      "2101831666542008740",
      "-117134843478280372887383049170"
    ],
    [
      "-2169848214851296298748860/175561",
      "13008881676193256049199642803123625370/73560059"
    ],
    [
      "34083123980688505600",
      "101424765412642883633658845170"
    ],
    [
      "14456812184419951300",
      "-10693854414303986395353522770"
    ],
    [
      "28837090862505308260",
      "41600529859801931766300221230"
    ],
    [
      "-29486747983714112060",
      "-174821473189313904124584652370"
    ],
    [
      "-5134582430554490960",
      "-152674321856149091526304523570"
    ],
    [
      "-26021305723258342620",
      "-184173321059307196694626320210"
    ],
    [
      "-36435628503899962460",
      "-131060345099252869970672533970"
    ],
    [
      "52968637037810831192260/4489",
      "14454982442327078126470423988704090/300763"
    ],
    [
      "27289337547927589864390/841",
      "2048997122207701765655443771878920/24389"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -2785/864 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -123856789139706169159947T^8 - 444860715403915335809784T^7 + 386197275608827081416504T^6 + 1527252937315431153244272T^5 - 1375436505233375995520880T^4 + 1122976665942695182127232T^3 - 920913248897305236746496T^2 - 558254638916475632295264T - 66664274811511860464112,\nA6 = 16359416402689331569871278519529286T^12 + 84307064804680497817942655822018088T^11 + 43582982500411768342190410573462056T^10 - 406313301062273227318175674455353520T^9 - 696574825662812869066591591106450400T^8 + 831815699985675007793533243030026816T^7 + 2339654226324227927870456252492273088T^6 - 2418192605459673378876521710121559264T^5 - 588423094738870081007533655250779760T^4 + 462827873819015983743769059572260800T^3 + 312326563016748800922506015981603904T^2 + 77162055820545612135770412695494272T + 6735862148389763480692013144006784.\nIt is a 2-neighbour of another fibration built here (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 04:09:39",
  "updated_at": "2026-10-14 04:09:39"
}