{
  "id": 5051,
  "curve_key": "205086819597036432159768889777200:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-4272642074938259003328518537025",
    "0"
  ],
  "rank_lower_bound": 7,
  "torsion": [
    2,
    2
  ],
  "family": "congruent",
  "families": [
    "quartic",
    "congruent"
  ],
  "naive_height": 223.20295856860224,
  "faltings_height": 16.66848695902501,
  "conductor": "136724546398024288106512593184800",
  "bad_primes": [
    "2",
    "3",
    "5",
    "11",
    "17",
    "73",
    "179",
    "1297",
    "43481"
  ],
  "discriminant": "4991941792245878152480048261519832834225175914229232248692954732535076779123709929787641000000",
  "regulator": "1531268431.219763467375827676714930498082693881088334",
  "points": [
    [
      "41943099540089002835481/18386944",
      "3633283914844497926070439023144579/78843215872"
    ],
    [
      "71118835921918072561/144",
      "599754003200335557946340391241/1728"
    ],
    [
      "-78366968545187523501255/47416996",
      "16478767274363771280225220718322075/326513434456"
    ],
    [
      "1218790703104598450409/183184",
      "40443914526059155144260927459573/78402752"
    ],
    [
      "-427424277885894167698655/239444676",
      "163149888155500767095654296221069175/3705166916424"
    ],
    [
      "132064993737026798333820/8334769",
      "47583278079684180423271271196497550/24062478103"
    ],
    [
      "904822407628307710792678780/216950076841",
      "23639371101720281671989146766133465393250/101050789840924139"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Congruent-number curve y^2 = x^3 - n^2 x with n = 2067037027955295, 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 01:50:10",
  "updated_at": "2026-10-21 01:50:10"
}