{
  "id": 1020,
  "curve_key": "3121:-64169",
  "ainvs": [
    "1",
    "1",
    "1",
    "-65",
    "47"
  ],
  "rank_lower_bound": 0,
  "torsion": [
    4,
    2
  ],
  "naive_height": 24.137726226812337,
  "faltings_height": 0.07092609103923017,
  "conductor": "390",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13"
  ],
  "discriminant": "15210000",
  "regulator": "1",
  "points": [],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 390b2 = LMFDB 390.f5: https://www.lmfdb.org/EllipticCurve/Q/390/f/5\n\nMordell–Weil rank exactly 0 (Cremona's tables). Conductor 390, minimal discriminant 15210000, torsion ℤ/2ℤ × ℤ/4ℤ.",
  "created_at": "2026-09-23 15:16:30",
  "updated_at": "2026-09-23 15:16:30"
}