{
  "id": 5052,
  "curve_key": "234618784742045357498206195393728:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-4887891348792611614545962404036",
    "0"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2,
    2
  ],
  "family": "congruent",
  "families": [
    "quartic",
    "congruent"
  ],
  "naive_height": 223.60654438259826,
  "faltings_height": 16.702119110191347,
  "conductor": "78206261580681785832735398464576",
  "bad_primes": [
    "2",
    "7",
    "17",
    "31",
    "41",
    "71",
    "1609",
    "63977"
  ],
  "discriminant": "7473853912566332254632952684780655154177148553190345353200947212874875063531485923913907113984",
  "regulator": "2536671208.046682482901675556086640626215374186570069",
  "points": [
    [
      "5021263312917826434649/746496",
      "336042530229492048500034915277475/644972544"
    ],
    [
      "40249934287301444434/6561",
      "238198027584946084809727121360/531441"
    ],
    [
      "27399556163232609567058/10962721",
      "2115283342657849841858487498436848/36297569231"
    ],
    [
      "757322711808724936478674/3783025",
      "659014523635375563427373218432699968/7357983625"
    ],
    [
      "686026594953575253436153522/23694137041",
      "17916022537587333950941018695592337728464/3647214820584089"
    ],
    [
      "129859639632476651429077457/16699359076",
      "1418760872087327205473578355272370243079/2157991375955176"
    ],
    [
      "30334927397988297085228801/12063946896",
      "79591708516753007056380017355020763265/1325055671269056"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Congruent-number curve y^2 = x^3 - n^2 x with n = 2210857604820494, rank 7 example from Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014, doi:10.5802/pmb.9; points from https://magma.maths.usyd.edu.au/~watkins/PTS.r6r7.",
  "created_at": "2026-10-21 01:50:15",
  "updated_at": "2026-10-21 01:50:15"
}