{
  "id": 5048,
  "curve_key": "12419321832636543408:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-258735871513261321",
    "0"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2,
    2
  ],
  "family": "congruent",
  "families": [
    "quartic",
    "congruent"
  ],
  "naive_height": 131.8973554381802,
  "faltings_height": 9.059686698156506,
  "conductor": "8279547888424362272",
  "bad_primes": [
    "2",
    "73",
    "137",
    "181",
    "281"
  ],
  "discriminant": "1108536267458170159577637191328154585728313173515466304",
  "regulator": "860240.250317562896676373022999180249673507959898",
  "points": [
    [
      "-470920189",
      "4172522207280"
    ],
    [
      "-72148036",
      "4276883208570"
    ],
    [
      "-31685193536863725/1302415921",
      "117790133059660800841584960/47002888172969"
    ],
    [
      "4588705434949/6241",
      "7097661704426143620/493039"
    ],
    [
      "480070559985773/779689",
      "5927121344625790341468/688465387"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Congruent-number curve y^2 = x^3 - n^2 x with n = 508660861. Found by a Mestre-Nagao sieve over n <= 3*10^9 (n = 5,6,7 mod 8), top candidates checked with ellrank.",
  "created_at": "2026-10-21 01:49:52",
  "updated_at": "2026-10-21 01:49:52"
}