{
  "id": 3355,
  "curve_key": "51664993:74487956815",
  "ainvs": [
    "1",
    "1",
    "1",
    "-1076354",
    "-86661394"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    4,
    2
  ],
  "naive_height": 53.28087297642703,
  "faltings_height": 2.505822629533817,
  "conductor": "5657379",
  "bad_primes": [
    "3",
    "7",
    "13",
    "17",
    "23",
    "53"
  ],
  "discriminant": "76596896839955187969",
  "regulator": "16.035375261801572831898021866819509534952328",
  "points": [
    [
      "-982",
      "5446"
    ],
    [
      "-948",
      "9577"
    ],
    [
      "7263",
      "608980"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/4ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with b=t^2-1/16, c=0, at t = -159/92; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 5657379, torsion ℤ/2ℤ × ℤ/4ℤ.",
  "created_at": "2026-09-26 21:11:06",
  "updated_at": "2026-09-26 21:11:06"
}