{
  "id": 6059,
  "curve_key": "976191376720:-964500485150963008",
  "ainvs": [
    "0",
    "1",
    "0",
    "-20337320348",
    "1116313226854980"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 82.82077346052816,
  "faltings_height": 4.121099645410976,
  "conductor": "245273964",
  "bad_primes": [
    "2",
    "3",
    "13",
    "19",
    "83",
    "997"
  ],
  "discriminant": "970342998325638912",
  "regulator": "717.43641659455321435767628863496520402149633",
  "points": [
    [
      "2971981/36",
      "23631013/216"
    ],
    [
      "43555384/529",
      "1943994/12167"
    ],
    [
      "81513715264/990025",
      "4282236762/985074875"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained in my own rational-parameter search of the Kubert-Tate family for torsion 6, following the construction described by David Renshaw on curve #1192 and the t=1/6916 example submitted by Lewis Combes on curve #5970. In y^2+(1-c)*x*y-b*y=x^3-b*x^2, set c=t and b=t*(t+1), then specialize at t=-1/997. Clearing denominators with x=X/997^2 and y=Y/997^3 gives [998,996,993012,0,0]; the displayed equation is its global minimal model, with discriminant 970342998325638912. All three listed rational points were checked exactly and returned as an independent set, with rank bounds [3,3]. The rational torsion subgroup is Z/6Z.",
  "created_at": "2026-10-28 17:58:26",
  "updated_at": "2026-10-28 17:58:26"
}