{
  "id": 6054,
  "curve_key": "96840675485569:-952986426827988071233",
  "ainvs": [
    "1",
    "0",
    "0",
    "-2017514072616",
    "1102993381443257920"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 96.61226466974531,
  "faltings_height": 5.152001122216462,
  "conductor": "68918290",
  "bad_primes": [
    "2",
    "5",
    "7",
    "157",
    "6271"
  ],
  "discriminant": "14876180743962890240",
  "regulator": "620.38654743917299254847042869891237917507939",
  "points": [
    [
      "7441336/9",
      "275146760/27"
    ],
    [
      "52491460784/64009",
      "-6640088404040/16194277"
    ],
    [
      "85556431184/104329",
      "-13817261415160/33698267"
    ]
  ],
  "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=1/6271. Clearing denominators with x=X/6271^2 and y=Y/6271^3 gives [6270,-6272,-39331712,0,0]; the displayed equation is its global minimal model, with discriminant 14876180743962890240. 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 17:38:14",
  "updated_at": "2026-10-28 17:38:14"
}