{
  "id": 1189,
  "curve_key": "3787798475521:-7335438536882711681",
  "ainvs": [
    "1",
    "1",
    "1",
    "-78912468240",
    "8490058018974705"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    8
  ],
  "naive_height": 86.88841826723142,
  "faltings_height": 5.076117134393857,
  "conductor": "779644110",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "23",
    "31",
    "41",
    "127"
  ],
  "discriminant": "310455332599837597122176100000000",
  "regulator": "72.940269891871439152821852606970462968352806",
  "points": [
    [
      "1553263/36",
      "15519728113/216"
    ],
    [
      "7438237/49",
      "1010404499/343"
    ],
    [
      "779400377/5329",
      "3389420987481/389017"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Found by a rational-parameter search of the Kubert-Tate family for torsion 8, at t=-10/21. In y^2+(1-c)xy-by=x^3-bx^2 the parameters are defined by b=(2*t-1)*(t-1);c=b/t; Converted to the global minimal model; PARI/GP ellrank returned bounds [3,3]. The 3 supplied points and the torsion subgroup are independently certified by the site.",
  "created_at": "2026-09-23 21:40:11",
  "updated_at": "2026-09-23 21:40:11"
}