{
  "id": 3279,
  "curve_key": "69648:-18380736",
  "ainvs": [
    "0",
    "0",
    "0",
    "-1451",
    "21274"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    4
  ],
  "naive_height": 33.453627791376235,
  "faltings_height": 0.3165110596186018,
  "conductor": "272",
  "bad_primes": [
    "2",
    "17"
  ],
  "discriminant": "69632",
  "regulator": "2.036297434690738342122104749035702383459",
  "points": [
    [
      "31",
      "78"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 272b3 = LMFDB 272.b1: https://www.lmfdb.org/EllipticCurve/Q/272/b/1\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 272, minimal discriminant 69632, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:50:38",
  "updated_at": "2026-09-26 20:50:38"
}