{
  "id": 4964,
  "curve_key": "2166816102711465617259003961:-100540260784937569774145429250983354753645",
  "ainvs": [
    "1",
    "1",
    "0",
    "-45142002139822200359562582",
    "116366042575140415108073024605154991000"
  ],
  "rank_lower_bound": 15,
  "torsion": [
    2
  ],
  "family": null,
  "families": [],
  "naive_height": 188.82916910152971,
  "faltings_height": 13.553569550376956,
  "conductor": "92600993893266656474637251754052208517553081703634",
  "bad_primes": [
    "2",
    "3",
    "19",
    "71",
    "73",
    "97",
    "127",
    "149",
    "151",
    "157",
    "199",
    "271",
    "463",
    "709",
    "941",
    "1217",
    "1423",
    "1783",
    "2087",
    "4481",
    "7487"
  ],
  "discriminant": "37648703923409297505266474057003443585161429152654447392536053614465261760652",
  "regulator": "4019888097853621891358058.9224626121604495528778080863551480784506498",
  "points": [
    [
      "530808882926005077299443259/8437357612369",
      "-12162392249684149635690969432091944679942/24508102342297195097"
    ],
    [
      "205955181000008751228539/14462227081",
      "-84519425695387006366358568429625082/1739212966533979"
    ],
    [
      "213095945307353624248379/47829252601",
      "-20071435035751175198237941143609114/10460209714586099"
    ],
    [
      "225949293199995633399659/55438586116",
      "3770435282342099452083370378350471/13053236855356664"
    ],
    [
      "201964724071905218445419/36956986564",
      "40737563314344450555434953838192565/7104685011036488"
    ],
    [
      "1793750091567875487341251/269874016036",
      "1470112163213227294269348699208863035/140197932086605784"
    ],
    [
      "1797977904653330255041171/217186757089",
      "1782126525738882390263773856835039643/101216195966457937"
    ],
    [
      "1913998953054441557676881/465027161041",
      "-171345508141836000741069190307316909/317115506901528089"
    ],
    [
      "10232904293913410445848148871456443454001/2247704276022060634448511241",
      "243248568643737530313802023608396843015629504646568917973604/106563569078924504416720363630290857133211"
    ],
    [
      "5668042088369813779999361/848393892889",
      "8305939312501670542535956808109137793/781441192043878787"
    ],
    [
      "34198834982015592864029832601/4943565707145025",
      "4040739699508700412015747445425233401916789/347584551934391744537375"
    ],
    [
      "378219541745131460305621336291/87233796880249761",
      "37757721057391518169844170222326941432593019/25764823016188083110720241"
    ],
    [
      "1523210661382349555024634096816401/370572119325925522201",
      "-3646713613350963252187031100819043474533127064255/7133605884838652004186897442051"
    ],
    [
      "79240510423460405373401701734780150533810990310580001/4562925384671156108502784859486961641209",
      "20835903862548398488684856675743754661745293724816858832596217252430847709640956/308223005197318164490945301157374328001584081389900184434573"
    ],
    [
      "1219314222512931389847912461279287116877639828969/191058782538103370684007717676084921",
      "784299576970045544983836651530565001441814245441513909934251141485265268/83512312370436683466988339519211764434007832342359469"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 2-isogeny from curve #4580, submitted by jesper-petersen: https://elliptic-rank.icarm.cloud/curve/4580.\r\n\r\nThe source commentary attributes the curve and independent-point witnesses to the Layr Labs Elliptic Rank Challenge public repository, submission/board/2-r15-z2k3.ec: https://github.com/Layr-Labs/elliptic-rank-challenge-public/blob/main/submission/board/2-r15-z2k3.ec.\r\n\r\nIt identifies the specialization u = 12/5, t = -159/38 of the Elkies–Klagsbrun Z/2Z family, citing https://arxiv.org/abs/2003.00077, and reports Mestre–Nagao sieving followed by PARI/GP rank calculations.\r\n\r\nApplied a rational degree-2 isogeny and converted the target to its minimal model y^2 + x*y = x^3 + x^2 - 45142002139822200359562582*x + 116366042575140415108073024605154991000.\r\n\r\nFifteen supplied rational points transported through this map and checked exactly on the target curve. Rational torsion Z/2Z. Same rank and conductor as the source; conductor 92600993893266656474637251754052208517553081703634. Source-attributed existing curve.",
  "created_at": "2026-10-20 23:58:58",
  "updated_at": "2026-10-20 23:58:58"
}