{
  "id": 6061,
  "curve_key": "291830120505:-157650434378829549",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-6079794177",
    "182467300479597"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 79.19831307246932,
  "faltings_height": 3.841413986454483,
  "conductor": "198793998",
  "bad_primes": [
    "2",
    "3",
    "13",
    "19",
    "61",
    "733"
  ],
  "discriminant": "156369588765041808",
  "regulator": "709.31571976170567256026953956883363974326976",
  "points": [
    [
      "720261/16",
      "-1400769/64"
    ],
    [
      "28710814/625",
      "4937783087/15625"
    ],
    [
      "2614690431/58081",
      "-315061688691/13997521"
    ]
  ],
  "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=1/732. Clearing denominators with x=X/732^2 and y=Y/732^3 gives [731,-733,-536556,0,0]; the displayed equation is its global minimal model, with discriminant 156369588765041808. 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 18:04:33",
  "updated_at": "2026-10-28 18:04:33"
}