{
  "id": 5769,
  "curve_key": "8175782848348735216741919296:-735063882732266654634712901298108208118656",
  "ainvs": [
    "0",
    "1",
    "0",
    "-170328809340598650348789985",
    "850768382791918518446322695470842310883"
  ],
  "rank_lower_bound": 19,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 192.8129219529109,
  "faltings_height": 13.90878630588223,
  "conductor": "1157495895621988921886541750950279165557035988363386565080",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "19",
    "53",
    "20759",
    "5992548933505173065457633575064363758180685809"
  ],
  "discriminant": "3575473056427564199341835438246690797565584909378977645054583313217468948000000",
  "regulator": "73343640904343016.06930944329338280023406037245330134237559852909851437426944",
  "points": [
    [
      "6254664545921",
      "5486936977884208770"
    ],
    [
      "-2899310492749",
      "-36335009000448628470"
    ],
    [
      "-12398502019189",
      "32506266694971172950"
    ],
    [
      "8014378092281",
      "674138656074042390"
    ],
    [
      "91491762917599/9",
      "-351832104162965606950/27"
    ],
    [
      "11732310440411",
      "-21617967529653575850"
    ],
    [
      "83334754626761",
      "-751924047586589758650"
    ],
    [
      "10391376232886",
      "-14243795890277008275"
    ],
    [
      "11621387636111",
      "-20996593846965209550"
    ],
    [
      "6939528380411",
      "1718726362846104150"
    ],
    [
      "5012629467791",
      "-11087054546998627890"
    ],
    [
      "8620637840177",
      "4803282756014094990"
    ],
    [
      "10185386935361",
      "-13136097401352144450"
    ],
    [
      "6884170590011",
      "-2109282130556404650"
    ],
    [
      "653377451441",
      "27198498260753383410"
    ],
    [
      "1792894860086",
      "23476582934707777875"
    ],
    [
      "18706295594561",
      "-64887264313801199550"
    ],
    [
      "6687035579246",
      "-3285427243228816455"
    ],
    [
      "12668329382227",
      "26945939296347441930"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -63/2 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -11858511001583521804032T^8 + 214175721852502381272960T^7 - 1432812052406818812164400T^6 + 1877396101235517581961840T^5 - 1616349363784284403162440T^4 + 375398756942628652907040T^3 + 1483849947541484408156640T^2 - 499683202991392785978600T - 424387235571887878825275,\nA6 = 508635860698696309787313618886656T^12 - 13266381990941392781192177963243520T^11 + 139080387761026008655151155338259200T^10 - 731079762012883508969321984907406080T^9 + 2344562178683608195181185875640917840T^8 - 1913494729584525923388480523171587648T^7 - 1946649819060578533441792459632712560T^6 + 2240017571252714640177648881635252560T^5 + 1187496071063136453053032076443570200T^4 - 1140089711639607323530481433639140640T^3 - 523526754044213182839802197400957584T^2 + 245269758021262210799450076829606200T + 113368132470196897298368287244953750.\nIt is a 2-neighbour of the fibration given in curve#4647: 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": "2026-10-24 22:57:39",
  "updated_at": "2026-10-24 22:57:39"
}