{
  "id": 1128,
  "curve_key": "2449:-121177",
  "ainvs": [
    "1",
    "0",
    "0",
    "-51",
    "136"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2
  ],
  "naive_height": 23.410305170856503,
  "faltings_height": -0.3417327711783586,
  "conductor": "2465",
  "bad_primes": [
    "5",
    "17",
    "29"
  ],
  "discriminant": "2465",
  "regulator": "1.71777543023359452600715033818409177601626",
  "points": [
    [
      "3",
      "2"
    ],
    [
      "8",
      "12"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cremona 2465a1, rank exactly 2 according to Cremona's tables. Witness generators from https://github.com/JohnCremona/ecdata/blob/master/allgens/allgens.00000-09999 . Found by screening the local allgens tables for missing torsion size records; torsion and independence independently verified by the site.",
  "created_at": "2026-09-23 21:36:54",
  "updated_at": "2026-09-23 21:36:54"
}