{
  "id": 6076,
  "curve_key": "25516097955176788245565109661376:-129410354332510265264441964137612485509587668480",
  "ainvs": [
    "0",
    "1",
    "0",
    "-531585374066183088449273117945",
    "149780502699664481675757214209355338100613079"
  ],
  "rank_lower_bound": 22,
  "torsion": [],
  "family": null,
  "families": [],
  "naive_height": 216.95863516324633,
  "faltings_height": 15.90671442377582,
  "conductor": "45002340928389264935278978544984321343300551847040480758707689509229958075325992064",
  "bad_primes": [
    "2",
    "3",
    "37",
    "593",
    "57193",
    "439133",
    "563490643",
    "377417299226393189022226470203914931977480653367934494393"
  ],
  "discriminant": "-77686281079132393158353912850807174373942130238094264800295840584477418186609150972305408",
  "regulator": "120238196861020554979022.9521382970321108861386314734497909359340681190137637596123",
  "points": [
    [
      "406709564503891",
      "-924425031209462566956"
    ],
    [
      "448717251669831",
      "1263694681466497151244"
    ],
    [
      "441518855834647",
      "-1069923676996867227036"
    ],
    [
      "442340979262042",
      "1090593967063921940421"
    ],
    [
      "646581113268229",
      "8739665552944955202348"
    ],
    [
      "306518407244755",
      "3954527546599460849748"
    ],
    [
      "367198550327875",
      "2023427293998970110948"
    ],
    [
      "343172938202743",
      "2787375786600550582980"
    ],
    [
      "140916076920679",
      "8813046907412868061548"
    ],
    [
      "439556531236515",
      "1022434580410704743748"
    ],
    [
      "223415293460674",
      "6493676862297759510393"
    ],
    [
      "437754375411939",
      "981359132772404107908"
    ],
    [
      "-7429940721988271/9",
      "-137654894416670062446088/27"
    ],
    [
      "1774255270856197/4",
      "-8976913968456693807945/8"
    ],
    [
      "3413147732864479/9",
      "44575584776541553141412/27"
    ],
    [
      "883449132433435450/1849",
      "175403967743049215740992489/79507"
    ],
    [
      "4716434081478961/25",
      "937146810351142262964384/125"
    ],
    [
      "472801039110643",
      "-2033892491398117404660"
    ],
    [
      "951957629167717",
      "-22503776591192606603196"
    ],
    [
      "22003611845029",
      "-11751355569102556578852"
    ],
    [
      "185579962923267",
      "-7584211775541544405596"
    ],
    [
      "4420962264312898/9",
      "-72371645615242787571397/27"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -55/13 of a rank-17 fibration of X1008 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -2350252713284250372T^8 - 50405464050571118556T^7 - 952448555730502901313T^6 - 695049490520215525224T^5 - 2931749924460790132683T^4 - 2386411337210880522384T^3 + 2409436881372804430248T^2 + 4544024515422355802304T - 2684309556378273707952,\nA6 = 1416965056476304965893460864T^12 + 45023054173800546171881279808T^11 + 917608982150765091196202404680T^10 + 4283108867178449968605589545480T^9 + 53149163760305922465025229930778T^8 + 64588783908002903981500312561392T^7 - 30251100417436562854623526202130T^6 - 127076856645436090494117187739472T^5 + 56548211197412673092440132446600T^4 + 156704525825325938035598980257408T^3 - 11739117611564936890190411272416T^2 - 136493880790478548113078471588096T + 53534237645294085780370164274560.\nIt is a 2-neighbour of the fibration given in curve#3589: 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": "2026-10-28 23:12:41",
  "updated_at": "2026-10-28 23:12:41"
}