{
  "id": 3365,
  "curve_key": "10267207056269881:136534919893813846821779",
  "ainvs": [
    "1",
    "0",
    "1",
    "-213900147005623",
    "-158026545479889387994"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6,
    2
  ],
  "naive_height": 110.60319429056362,
  "faltings_height": 7.284118603124446,
  "conductor": "23094404970",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "41",
    "53",
    "229"
  ],
  "discriminant": "615556365357650147808428076523365852365062500",
  "regulator": "112.02231211828037997631402219144849096163569",
  "points": [
    [
      "-51170372870/4489",
      "8503560453346376/300763"
    ],
    [
      "-1890438050/841",
      "430429347854308/24389"
    ],
    [
      "-1413837695/169",
      "71061873398294/2197"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found in a rational-parameter search of the Kubert–Tate family for torsion ℤ/2ℤ × ℤ/6ℤ: y^2 + (1-c)xy - by = x^3 - bx^2 with c=(10-2t)/(t^2-9), b=c+c^2, at t = -89/41; converted to the global minimal model. The 3 witness points were found with PARI/GP ellrank. Conductor 23094404970, torsion ℤ/2ℤ × ℤ/6ℤ.",
  "created_at": "2026-09-26 21:12:43",
  "updated_at": "2026-09-26 21:12:43"
}