{
  "id": 5970,
  "curve_key": "2291780948089369:-109713395742620851004797",
  "ainvs": [
    "1",
    "0",
    "1",
    "-47745436418529",
    "126983092945543320452"
  ],
  "rank_lower_bound": 4,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 106.10431685042703,
  "faltings_height": 5.9374419233526865,
  "conductor": "33127795610",
  "bad_primes": [
    "2",
    "5",
    "7",
    "13",
    "19",
    "277",
    "6917"
  ],
  "discriminant": "109618302873088494976400",
  "regulator": "5499.680356228010328979227066052181515662194034",
  "points": [
    [
      "319505392567/80089",
      "-45192584922242/22665187"
    ],
    [
      "55057591",
      "405430756254"
    ],
    [
      "3661308426359/917764",
      "-1753759063162297/879217912"
    ],
    [
      "145582140976/36481",
      "16233077472629/6967871"
    ]
  ],
  "submitter": "Lewis Combes",
  "history": [],
  "commentary": "Found by a rational-parameter search of the Kubert-Tate family for torsion 6, at t=1/6916. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by c=t; b=t*(t+1); converted to the global minimal model. \r\nTorsion Z/6, rank exactly 4 (GRH): the four points are independent by\r\nthe mod-2 Kummer-map test and the 2-Selmer rank is 4. \r\nConductor 33127795610 = 2 * 5 * 7 * 13 * 19 * 277 * 6917, \r\nminimal discriminant 2^4 * 5^2 * 7^2 * 13^2 * 19^2 * 277 * 6917^3 = 109618302873088494976400.",
  "created_at": "2026-10-27 07:19:08",
  "updated_at": "2026-10-27 07:19:08"
}