{
  "id": 3293,
  "curve_key": "14641:-3999689",
  "ainvs": [
    "1",
    "1",
    "1",
    "-305",
    "4502"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    4
  ],
  "naive_height": 30.403454332122955,
  "faltings_height": 0.5824226393759269,
  "conductor": "57855",
  "bad_primes": [
    "3",
    "5",
    "7",
    "19",
    "29"
  ],
  "discriminant": "-7441599375",
  "regulator": "2.59679098815781680378950072846032755807474",
  "points": [
    [
      "2",
      "61"
    ],
    [
      "5",
      "53"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 57855i1 = LMFDB 57855.d4: https://www.lmfdb.org/EllipticCurve/Q/57855/d/4\n\nMordell–Weil rank exactly 2 (Cremona's tables). Conductor 57855, minimal discriminant -7441599375, torsion ℤ/4ℤ. The witnesses are the generators from Cremona's tables.",
  "created_at": "2026-09-26 20:52:53",
  "updated_at": "2026-09-26 20:52:53"
}