{
  "id": 6057,
  "curve_key": "1267241021881:-1426556032247930221",
  "ainvs": [
    "1",
    "0",
    "1",
    "-26400854623",
    "1651104318734256"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 83.60358968864338,
  "faltings_height": 4.134103420497258,
  "conductor": "149368310",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "19",
    "53",
    "163"
  ],
  "discriminant": "22227016498300250",
  "regulator": "617.42763414833513427023285909069459180880724",
  "points": [
    [
      "13575685/144",
      "345935171/1728"
    ],
    [
      "7598896/81",
      "-32733289/729"
    ],
    [
      "4623343435/49284",
      "-510404902139/10941048"
    ]
  ],
  "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=1/2119. Clearing denominators with x=X/2119^2 and y=Y/2119^3 gives [2118,-2120,-4492280,0,0]; the displayed equation is its global minimal model, with discriminant 22227016498300250. 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 17:45:19",
  "updated_at": "2026-10-28 17:45:19"
}