{
  "id": 2929,
  "curve_key": "3808:31616",
  "ainvs": [
    "0",
    "1",
    "0",
    "-79",
    "-63"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2
  ],
  "naive_height": 24.734578187733767,
  "faltings_height": 0.1283463766784146,
  "conductor": "214912",
  "bad_primes": [
    "2",
    "23",
    "73"
  ],
  "discriminant": "31377152",
  "regulator": "6.3153346861019614796392101114828234018194451",
  "points": [
    [
      "-1",
      "4"
    ],
    [
      "-7",
      "14"
    ],
    [
      "9",
      "6"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 214912a1 = LMFDB 214912.a2: https://www.lmfdb.org/EllipticCurve/Q/214912/a/2\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 214912, minimal discriminant 31377152, torsion ℤ/2ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:28:49",
  "updated_at": "2026-09-26 20:28:49"
}