{
  "id": 4207,
  "curve_key": "6045734928186249500964429038475703348416:-320103798921532481557299043392327423611677923400152165169536",
  "ainvs": [
    "0",
    "0",
    "0",
    "-125952811003880197936758938301577153092",
    "370490508011032964765392411333712295846849448379805746724"
  ],
  "rank_lower_bound": 25,
  "torsion": [],
  "naive_height": 274.80051503896266,
  "faltings_height": 20.931345445583887,
  "conductor": "2298247654048435327349839964710891160783031039915487120729729326813369236867188725107851658520",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "23",
    "41",
    "47",
    "59",
    "79",
    "107",
    "4079183551567",
    "64903365949703",
    "6495083051416390583",
    "11541019832122197760104771350479"
  ],
  "discriminant": "68582567204242752608347261598231535636763609884677142368780161034077040120811678063349721022871229834170502772000000",
  "regulator": "62681866640638169886891645.09203701394498239065179047258539158560021160053615933029261532",
  "points": [
    [
      "1904134729006247808",
      "11728736632541369575110495750"
    ],
    [
      "9281692896433979553",
      "-1025376622840593668450332065"
    ],
    [
      "43434261185653795758",
      "277200431253422566671921065850"
    ],
    [
      "9835760512847403358",
      "9120404864335262973377829850"
    ],
    [
      "79435266124724243958",
      "-701141117591925513699510340950"
    ],
    [
      "14415377173114075558",
      "-39375033737445445727139685750"
    ],
    [
      "2126951864886336558",
      "-10593257801453118259777328250"
    ],
    [
      "-10757931779222429592",
      "21918793375450906526760022050"
    ],
    [
      "10542646251048637488",
      "14642479008114564153776993610"
    ],
    [
      "1857508623772637598",
      "11955798112492517659868075430"
    ],
    [
      "-3752828423254701567",
      "28112559613218846044059328625"
    ],
    [
      "3061000223853438288",
      "-3691835262646360109801288790"
    ],
    [
      "313106047273401643002/169",
      "-26317941382020644364201642475950/2197"
    ],
    [
      "-6360644248979103552",
      "-30237293003930591949315276870"
    ],
    [
      "2870483322234641128",
      "-5709370620127827517502365090"
    ],
    [
      "9478158041358008688",
      "5307045823097472068087732490"
    ],
    [
      "-69083457431291999682/121",
      "-27989496538465354158561827881650/1331"
    ],
    [
      "26227361579332674600",
      "-122915473819524218721648901218"
    ],
    [
      "24411716228090874918",
      "108827783208585269391251304330"
    ],
    [
      "1905123211728872920",
      "-11723886655547986140149022578"
    ],
    [
      "-1245725603931842221482/121",
      "31943714644642417420138592706750/1331"
    ],
    [
      "13912869581483144034",
      "-36210647320214883807567051930"
    ],
    [
      "-8979155760934921422",
      "-27883576611862023119298770490"
    ],
    [
      "2486402305437461564913/256",
      "32609194514991975573932198960055/4096"
    ],
    [
      "-4007784356536177455",
      "28476443543290699201876487703"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -148/49 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -37778694267449139654832T^8 - 165004421248548489849920T^7 + 23964366446631678311360T^6 + 575308331009960686683200T^5 + 644216561885971678915960T^4 + 812460692539323896913800T^3 - 3819953879838341796444940T^2 - 3397350178704392175665600T - 728947575597452632980100,\nA6 = 2901058156988098667187936175477120T^12 + 21730259143975499723992766217644800T^11 + 17535187883067791568400114163932160T^10 - 192962391515084787078631234218208000T^9 - 223683722196314018130030826199045280T^8 + 1043994033704842065087271937128639200T^7 + 875242041280405478572518252943970640T^6 - 3146689238453170755654295705800631200T^5 - 1529213271890985767861180186700707580T^4 + 3040571437100170751919897956070815600T^3 + 3919924516140400157697908339646928136T^2 + 1660827326525159620863953893304037040T + 238239533579990858915246998091788900.\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 22:51:09",
  "updated_at": "2026-10-13 22:51:09"
}