{
  "id": 2867,
  "curve_key": "721:-1081",
  "ainvs": [
    "1",
    "0",
    "0",
    "-15",
    "0"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "naive_height": 19.741917411854846,
  "faltings_height": -0.28678946503578956,
  "conductor": "465",
  "bad_primes": [
    "3",
    "5",
    "31"
  ],
  "discriminant": "216225",
  "regulator": "1.093567343958523129336386005379178616190",
  "points": [
    [
      "5",
      "5"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 465b2 = LMFDB 465.a2: https://www.lmfdb.org/EllipticCurve/Q/465/a/2\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 465, minimal discriminant 216225, torsion ℤ/2ℤ × ℤ/2ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:26:03",
  "updated_at": "2026-09-26 20:26:03"
}