{
  "id": 1031,
  "curve_key": "79729:-22667401",
  "ainvs": [
    "1",
    "0",
    "0",
    "-1661",
    "26097"
  ],
  "rank_lower_bound": 0,
  "torsion": [
    7
  ],
  "naive_height": 33.87287674201264,
  "faltings_height": 0.6714922026412216,
  "conductor": "678",
  "bad_primes": [
    "2",
    "3",
    "113"
  ],
  "discriminant": "-4048994304",
  "regulator": "1",
  "points": [],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 678d1 = LMFDB 678.e2: https://www.lmfdb.org/EllipticCurve/Q/678/e/2\n\nMordell–Weil rank exactly 0 (Cremona's tables). Conductor 678, minimal discriminant -4048994304, torsion ℤ/7ℤ.",
  "created_at": "2026-09-23 15:18:37",
  "updated_at": "2026-09-23 15:18:37"
}