{
  "id": 2831,
  "curve_key": "47281:-10093321",
  "ainvs": [
    "1",
    "0",
    "0",
    "-985",
    "11600"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    4,
    2
  ],
  "naive_height": 32.29159140734558,
  "faltings_height": 0.5828864099712382,
  "conductor": "3135",
  "bad_primes": [
    "3",
    "5",
    "11",
    "19"
  ],
  "discriminant": "2211350625",
  "regulator": "1.867959319870033007330448948518889446206",
  "points": [
    [
      "25",
      "40"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 3135e2 = LMFDB 3135.e3: https://www.lmfdb.org/EllipticCurve/Q/3135/e/3\n\nMordell–Weil rank exactly 1 (Cremona's tables). Conductor 3135, minimal discriminant 2211350625, torsion ℤ/2ℤ × ℤ/4ℤ. The witness is the generator from Cremona's tables.",
  "created_at": "2026-09-26 20:24:24",
  "updated_at": "2026-09-26 20:24:24"
}