{
  "id": 6369,
  "curve_key": "3828247558922496985:-7809467631745675646545351453",
  "ainvs": [
    "1",
    "0",
    "1",
    "-79755157477552021",
    "9038735678244565171483340"
  ],
  "rank_lower_bound": 6,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 128.45026861483632,
  "faltings_height": 8.633319519025607,
  "conductor": "604178875329281574",
  "bad_primes": [
    "2",
    "3",
    "7",
    "19",
    "43",
    "61",
    "103",
    "113",
    "3691",
    "6719"
  ],
  "discriminant": "-2825799513350230533414215720986625061342967542250428",
  "regulator": "227268.76450468031569766650103081264309462945361183",
  "points": [
    [
      "22876369",
      "2688147618980"
    ],
    [
      "451924261",
      "8080271520170"
    ],
    [
      "-449194432944/15625",
      "6567819294144196079/1953125"
    ],
    [
      "48272479786/625",
      "28553039693856266/15625"
    ],
    [
      "301774534681/961",
      "115219026863525030/29791"
    ],
    [
      "868811651951629/327184",
      "25469482478156999565295/187149248"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/6-torsion Kubert–Tate family curve from my rational-parameter search of y^2+(1-c)xy-by=x^3-bx^2, with b=c(c+1), following the construction of David Renshaw on curve#1192 and the example of Lewis Combes on curve#5970.\r\n\r\nThe completed rank-6 batch identified the integral model [48077,138942201,3806738422998,0,0], which reduces to the displayed global minimal model.\r\n\r\nA separate computation with PARI/GP 2.19.0 returned rank bounds [6,6] and the six displayed rational points. All six points were checked exactly on this model. The minimal discriminant was fully factored and every listed bad-reduction prime proved prime. The conductor is 604178875329281574.",
  "created_at": "2026-11-01 16:20:01",
  "updated_at": "2026-11-01 16:20:01"
}