{
  "id": 4731,
  "curve_key": "2191878278692644184516800:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-45664130806096753844100",
    "0"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    2,
    2
  ],
  "family": "congruent",
  "families": [
    "quartic",
    "congruent"
  ],
  "naive_height": 168.1404032088226,
  "faltings_height": 12.079940679043373,
  "conductor": "730626092897548061505600",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "103",
    "1249",
    "3257"
  ],
  "discriminant": "6094043647248539371477918240381094056566185517719091569350407744000000",
  "regulator": "9537625.0794584170331012754066664824450602457304362",
  "points": [
    [
      "149669647271916/49",
      "1826565232243870153764/343"
    ],
    [
      "-135022380906",
      "60861159901278072"
    ],
    [
      "45030121627046640/157609",
      "6342661252892826238293000/62570773"
    ],
    [
      "1157482044084685/441",
      "39248875014125135685425/9261"
    ],
    [
      "2566032106948232640/11336689",
      "1355219411203866924163285200/38170631863"
    ],
    [
      "-1601233843245878531514/83037914569",
      "707157180081032939686674114839784/23928454575946747"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Congruent-number curve y^2 = x^3 - n^2 x with n = 213691672290. Source: Watkins et al. electronic data http://magma.maths.usyd.edu.au/~watkins/PTS.r6r7 (paper Sec. 15); Watkins, Donnelly, Elkies, Fisher, Granville, Rogers, 'Ranks of quadratic twists of elliptic curves', Publ. Math. Besancon 2014/2, 63-98, doi:10.5802/pmb.9 (https://pmb.centre-mersenne.org/item/10.5802/pmb.9.pdf).",
  "created_at": "2026-10-20 21:26:38",
  "updated_at": "2026-10-20 21:26:38"
}