{
  "id": 4228,
  "curve_key": "3828214557633691420415663185564900875201042001:-236880574949325341524101670424011852018385955069988664148095583063001",
  "ainvs": [
    "1",
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    "0",
    "-79754469950701904591992983032602101566688375",
    "274167332117274700838073990858999307269971448582419348144424515625"
  ],
  "rank_lower_bound": 26,
  "torsion": [],
  "naive_height": 314.87634449714574,
  "faltings_height": 23.92866119955007,
  "conductor": "445667523771971004518111248205490995553686570820602547907604057815518927174609351664843296976410",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "19",
    "29",
    "41",
    "47",
    "71",
    "38196596281056749",
    "563035424011806987299136618729228511340687494179367709500649345243"
  ],
  "discriminant": "-5239926344304762186443428115709005975086173540106654333605808735583772712823958674617131999977910168073777917732348913198080000000",
  "regulator": "295656944766532255760679154.795617733548791943236618737138028913495370028514326175428955664",
  "points": [
    [
      "5164056838845195992250",
      "-4807731343779708170848175215875"
    ],
    [
      "1490115136642206871830",
      "398287214371865304743010447294645"
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    [
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      "-231234705470648262357816364253775"
    ],
    [
      "4759325340314557303500",
      "48930422733741808967880096387375"
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    [
      "5084378491576216827390",
      "10060022897782300611406739269725"
    ],
    [
      "291625301548199881847750/49",
      "-34834018823598973356667951977770125/343"
    ],
    [
      "13311788895120363123410",
      "1253551444700482028832784931632865"
    ],
    [
      "5354968542014039719650",
      "-25338943689903513971287914053775"
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      "-8799409981140587210550227140875"
    ],
    [
      "2226197240089021929750",
      "328102244250521982207734696659125"
    ],
    [
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      "-4122097844735203741659983386870875"
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      "-40505666835243091224532100680875"
    ],
    [
      "5131266305028690374790",
      "-5621844235241745776693290093275"
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    [
      "5314087987232355300750",
      "20307993480414600768922498630125"
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    [
      "5266643393844334548750",
      "14583307266946348610112279862125"
    ],
    [
      "18198250041510009969302310/3721",
      "-7502858640138222500520522342208826775/226981"
    ],
    [
      "5585125370332388040150",
      "-54304025870110199797840418899275"
    ],
    [
      "38690146228588798129890",
      "-7423260457164135848430982863849615"
    ],
    [
      "3957955688305656792150",
      "-143198063933612787433253629540875"
    ],
    [
      "5132988311501315121450",
      "5507614241196722655978178714425"
    ],
    [
      "4983548064982514667750",
      "21846238436695742300872808479125"
    ],
    [
      "414692772875482621225750/81",
      "-4755251102440597496273828927157875/729"
    ],
    [
      "5134730089229800077750",
      "-5398222333920525782742965320875"
    ],
    [
      "3533647270693570912662",
      "190962377966770662130919230110261"
    ],
    [
      "-7330761085537567123050",
      "-681815789066346295443387197616075"
    ],
    [
      "5177435654466304087650",
      "5404380468876026433450358615725"
    ]
  ],
  "submitter": "apkorte",
  "history": [],
  "commentary": "Specialization at T = -18356/2459 of a rank-17 fibration of X1152 not posted before: y^2 = x^3 + A4 x + A6, where\nA4 = -10669947921935414766592T^8 + 11909462495223346810112T^7 + 29290850738695015599056T^6 + 749826724411797003953776T^5 - 2800296645908250177432680T^4 + 5291011042236720751525024T^3 - 9053743075112346775820704T^2 - 1373686135035664689321352T - 153062929487822447776267,\nA6 = 424084459520579325699737587892224T^12 - 702971076900016054991835746922496T^11 - 1428460164091780099364248773693184T^10 - 45531551300070380060674898437602560T^9 + 178145138683306119166957105019549520T^8 - 224307849011465835658368095576866752T^7 + 1890363249575504635685922631864051728T^6 - 8402714629932209203714457518206796848T^5 + 10932033646871752694840038266436614840T^4 - 1045636958253350245306227479236779680T^3 + 3911537622199302231442442603656239344T^2 + 397823992391227625022128206594583384T + 16543056563137220502197778370669126.\nIt is a 2-neighbour of the fibration given in curve#4205 (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 19:06:09",
  "updated_at": "2026-10-14 19:06:09"
}