{
  "id": 6052,
  "curve_key": "1861074324736:-2538899888356933984",
  "ainvs": [
    "0",
    "1",
    "0",
    "-38772381765",
    "2938528613322900"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 84.75652509256369,
  "faltings_height": 4.27084808557578,
  "conductor": "530673780",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "83",
    "1171"
  ],
  "discriminant": "2551981392433746000",
  "regulator": "276.01480865152736270858662984662485228197885",
  "points": [
    [
      "119613",
      "3492489"
    ],
    [
      "125565",
      "7058295"
    ],
    [
      "95608365/841",
      "2073555/24389"
    ]
  ],
  "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=-1/1171. Clearing denominators with x=X/1171^2 and y=Y/1171^3 gives [1172,1170,1370070,0,0]; the displayed equation is its global minimal model, with discriminant 2551981392433746000. 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 17:33:23",
  "updated_at": "2026-10-28 17:33:23"
}