{
  "id": 4210,
  "curve_key": "157301247194642625915478224371281:-1979708895964878640335456992147873024284351571929",
  "ainvs": [
    "1",
    "0",
    "0",
    "-3277109316555054706572463007735",
    "2291329740700090745814335879842516378549785897"
  ],
  "rank_lower_bound": 22,
  "torsion": [],
  "naive_height": 222.4140685507437,
  "faltings_height": 16.355283262375266,
  "conductor": "1667588633554212862622195780664243944818507616821531290807206649479430",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "17",
    "23",
    "1123",
    "98939",
    "101966951",
    "24892688260915228306151",
    "38776540063146823717824631"
  ],
  "discriminant": "-15645379590423656245471234481371659189365245334795514659333720849365071884669633414400000000",
  "regulator": "169534326506851740200.4234427081769779511286835726977126697101617868976148673279137",
  "points": [
    [
      "760236700819294",
      "15470528007536435358013"
    ],
    [
      "-1177430374063466",
      "67212918045313248231733"
    ],
    [
      "1233410735645488",
      "-11211521456136932164499"
    ],
    [
      "-450236928565466",
      "-60626201734594017134267"
    ],
    [
      "44144929213902526/49",
      "-2872958260883530445874221/343"
    ],
    [
      "1054194615046414",
      "-2858834720295117678587"
    ],
    [
      "2176253369730034",
      "73935219486611360687233"
    ],
    [
      "4245126330857734",
      "254718317744271342949333"
    ],
    [
      "1076517892674934",
      "3321080345367767734933"
    ],
    [
      "429517239232812454/361",
      "60004702978752564330689767/6859"
    ],
    [
      "-1669220109300266",
      "55772804036807487034933"
    ],
    [
      "1128374171544214",
      "-5495646664317883924427"
    ],
    [
      "1054056956913334",
      "-2857475691702991181867"
    ],
    [
      "-469215646844426",
      "61038485080811090290453"
    ],
    [
      "-756734955078716",
      "65862662872502069210983"
    ],
    [
      "1404192585640582",
      "21409423390462575542485"
    ],
    [
      "1448297580549934",
      "24145388857085764965133"
    ],
    [
      "2160990740328534",
      "72808627586259596327733"
    ],
    [
      "2412061824904534",
      "-91761741938535555824267"
    ],
    [
      "50276691482098558/49",
      "-1032118254009568978862765/343"
    ],
    [
      "977518531611034",
      "-4685631208837265799767"
    ],
    [
      "987604865243734",
      "-4256083808563803421067"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = 245/163 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -1383044527665321116400T^8 - 16049999528139513071520T^7 + 15805299603407933138816T^6 + 712593854642478382322880T^5 - 3813994880887522829708400T^4 + 10533902963478229407850960T^3 - 10115880489507421099979016T^2 - 8597064026478854771701800T - 1511313760202147775869947,\nA6 = 19606871590209659689292333520000T^12 + 323679316365878812163330339568000T^11 + 983969135059873668251937025883200T^10 - 6275524572874812809451065998804160T^9 - 65918284694373505756900522296861936T^8 - 337269555874810038655223788688478240T^7 + 5081060790854484290367532173908523840T^6 - 14611633213803604472414514215799717120T^5 + 11577647003766635578545334685110815264T^4 + 380100766090866777957899579121104880T^3 + 4610206606796227830325446909775634088T^2 + 8093183322260907342485075849728593400T + 2171157485142496238941816568266368214.\nIt is a 2-neighbour of the fibration given in curve#4191 (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 01:22:34",
  "updated_at": "2026-10-14 01:22:34"
}