{
  "id": 6370,
  "curve_key": "4066092956024905321:-22017955458040047117716083381",
  "ainvs": [
    "1",
    "0",
    "1",
    "-84710269917185528",
    "25483744736042717226480006"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 130.52331157701204,
  "faltings_height": 8.929132493441799,
  "conductor": "725155197682921070",
  "bad_primes": [
    "2",
    "5",
    "13",
    "31",
    "53",
    "79",
    "131",
    "421",
    "443",
    "1759"
  ],
  "discriminant": "-241646523498000986859943224700285024021673754751000000",
  "regulator": "82643.084387927289631061174537090836649723915080777",
  "points": [
    [
      "-229241343",
      "5732116274259"
    ],
    [
      "113843345",
      "4161131487907"
    ],
    [
      "-926752600357/9604",
      "5387040370112308291/941192"
    ],
    [
      "96764254640/361",
      "32197037014616623/6859"
    ],
    [
      "2771591849035/324",
      "4611570941071114559/5832"
    ],
    [
      "3880919764320/361",
      "7642670553078755798/6859"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/6-torsion Kubert–Tate family curve from a rational-parameter search of\r\ny^2+(1-c)xy-by=x^3-bx^2, with b=c(c+1), following the construction of\r\nDavid Renshaw on curve#1192 and the example of Lewis Combes on curve#5970.\r\n\r\nThe completed rank-6 batch identified the specialization with raw model\r\n[64089,476741095,20873632103480,0,0], reducing to the displayed global\r\nminimal model.\r\n\r\nPARI/GP returned rank bounds [6,6]. The six displayed points were checked\r\nexactly on the minimal model. The minimal discriminant was fully factored,\r\nand the listed bad-reduction primes are complete.",
  "created_at": "2026-11-01 16:24:32",
  "updated_at": "2026-11-01 16:24:32"
}