{
  "id": 869,
  "curve_key": "-32:8",
  "ainvs": [
    "0",
    "1",
    "1",
    "1",
    "0"
  ],
  "rank_lower_bound": 0,
  "torsion": [
    3
  ],
  "naive_height": 10.39720770839918,
  "faltings_height": -1.0651731316767918,
  "conductor": "19",
  "bad_primes": [
    "19"
  ],
  "discriminant": "-19",
  "regulator": "1",
  "points": [],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "LMFDB 19.a3 (Cremona 19a3): https://www.lmfdb.org/EllipticCurve/Q/19/a/3\n\nMordell–Weil rank exactly 0 per LMFDB. Conductor 19, minimal discriminant -19, torsion ℤ/3ℤ.\n\nSubmitted as one of the ten smallest rank-0 curves in LMFDB (as of 2026-09-23; places in LMFDB sort order, ties broken by label): #4 by Faltings height, #8 by |minimal discriminant|.",
  "created_at": "2026-09-23 14:40:24",
  "updated_at": "2026-09-23 14:40:24"
}