{
  "id": 6088,
  "curve_key": "12143031073623751181598413128099300625362195825:399445574308215317720860165904861192811035196961057386347557726624375",
  "ainvs": [
    "1",
    "1",
    "1",
    "-252979814033828149616633606835402096361712413",
    "-462321266560434395510360229904992327482519890115183022109083267469"
  ],
  "rank_lower_bound": 19,
  "torsion": [
    2
  ],
  "family": null,
  "families": [],
  "naive_height": 318.339253847324,
  "faltings_height": 24.590862296579516,
  "conductor": "85846100457142021413369807897710852813676285438536199952814698136832974679317780550",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "13",
    "19",
    "31",
    "71",
    "73",
    "79",
    "83",
    "157",
    "211",
    "223",
    "227",
    "251",
    "443",
    "521",
    "1627",
    "3191",
    "4373",
    "7109",
    "8731",
    "27061",
    "109829",
    "118189",
    "184181",
    "64459907"
  ],
  "discriminant": "943849575841156867304711452333072169478500186162127472030386330525094527564154663695214500770426151452281646039509369009284059136000000",
  "regulator": "8258772015982912791327.412174130226459595433105008054949092895994449335736544",
  "points": [
    [
      "-243951585533674722217441/25",
      "129729944187787006341175509767978292/125"
    ],
    [
      "-48412064267562967596019/25",
      "17813264318385822059421000562064364/125"
    ],
    [
      "436501928952635360289119/25",
      "83236406790167214406970792302999668/125"
    ],
    [
      "16750667626849762706885",
      "9043385904339763285878131763132"
    ],
    [
      "-361750815823269406888369/25",
      "51317452075945279545844860074449764/125"
    ],
    [
      "-2994521673691748102665",
      "-518053968650976491504821722926268"
    ],
    [
      "-127956732324076294260769/25",
      "-104464099820258133087115113075666636/125"
    ],
    [
      "18022254102948512989015",
      "912178701961024332858807375151492"
    ],
    [
      "31330983422122156969015",
      "-4729378605226572690024371666744508"
    ],
    [
      "-10014385309347949257865",
      "1032857329342665507322838624712132"
    ],
    [
      "-6360565789011542744201",
      "943104110888088791391807566734788"
    ],
    [
      "16762715767786094852855",
      "84750932834911395373189293743172"
    ],
    [
      "92807539700322117156651",
      "27846594127498834742892090274449688"
    ],
    [
      "1184613571403225581857271",
      "1289216472151705599979788214200579780"
    ],
    [
      "-15610181597903220709745844049/1776889",
      "2463924075442297697904939211514263943766756/2368593037"
    ],
    [
      "2097648368209488735969785255/14161",
      "95510177394116596249425380795101505827548/1685159"
    ],
    [
      "124931467769404027031132290135/3157729",
      "40251889445559018793070707339143351283567236/5611284433"
    ],
    [
      "207158437295224547487034990638815/5776152001",
      "2655659725938364344603279142258190716246510922892/438993328228001"
    ],
    [
      "45534635132210116682386961268107495/17697779089",
      "9716379901021380242405125118017550018891438666780964/2354388645546937"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Original specialization/section search by spariev (2026), public Yukon elliptic-rank challenge submission c2120023-a114-4514-ac1b-ff2e5824b97f. Source: https://raw.githubusercontent.com/Layr-Labs/elliptic-rank-challenge-public/7520ba9cfcdc31043e79e9baa221b5e59f321170/submission/board/2-19-x552-p18371-q22532.ec . Specialization T=18371/22532 of the rank-nine X552 fibration posted by wgxli with curve#4061: y^2+238(19T-6)(5T-3)xy=x(x+F)(x+G), F=(12T-29)(51T+22)(79T+22)(482T+29), G=256(T-22)(83T-7)(21T-2)(7T+17). The nineteen supplied points have been checked exactly on the minimal model and certified independent modulo torsion by an exact rank-19 F_5 reduction matrix at nineteen good primes.",
  "created_at": "2026-10-29 01:37:14",
  "updated_at": "2026-10-29 01:37:14"
}