{
  "id": 3285,
  "curve_key": "8416:-772064",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-175",
    "952"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    4
  ],
  "naive_height": 27.113669804932474,
  "faltings_height": -0.10680569706124192,
  "conductor": "840",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7"
  ],
  "discriminant": "8400",
  "regulator": "2.196597963753069286457897954426581558085",
  "points": [
    [
      "17",
      "51"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 840f1 = LMFDB 840.d4: https://www.lmfdb.org/EllipticCurve/Q/840/d/4\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 840, minimal discriminant 8400, torsion ℤ/4ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:51:35",
  "updated_at": "2026-09-26 20:51:35"
}