{
  "id": 2708,
  "curve_key": "31035408:-172856479680",
  "ainvs": [
    "0",
    "0",
    "0",
    "-646571",
    "200065370"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    4
  ],
  "naive_height": 51.751917912573106,
  "faltings_height": 2.035276869291805,
  "conductor": "449072",
  "bad_primes": [
    "2",
    "13",
    "17",
    "127"
  ],
  "discriminant": "8006561105813504",
  "regulator": "22.452312071989849225401070877668572098967919",
  "points": [
    [
      "445",
      "680"
    ],
    [
      "479",
      "510"
    ],
    [
      "-371",
      "19720"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 449072l3 = LMFDB 449072.l1: https://www.lmfdb.org/EllipticCurve/Q/449072/l/1\n\nMordell–Weil rank exactly 3 (Cremona's tables). Conductor 449072, minimal discriminant 8006561105813504, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:18:42",
  "updated_at": "2026-09-26 20:18:42"
}