{
  "id": 3473,
  "curve_key": "34987838017367248:-5172820284163692055636160",
  "ainvs": [
    "0",
    "-1",
    "0",
    "-728913292028484",
    "5987060757049444629444"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    4,
    2
  ],
  "naive_height": 114.28133073238513,
  "faltings_height": 7.534399885486683,
  "conductor": "38019384450552",
  "bad_primes": [
    "2",
    "3",
    "37",
    "47",
    "73",
    "97",
    "103",
    "1249"
  ],
  "discriminant": "9301070920556501722516656067077726090938513664",
  "regulator": "11891.0701626527169885580441754187722545755216233",
  "points": [
    [
      "-21809604",
      "107286718110"
    ],
    [
      "8828098",
      "15497202840"
    ],
    [
      "39632744",
      "198372206186"
    ],
    [
      "-9527640348/361",
      "567311611376910/6859"
    ],
    [
      "7684024869/3025",
      "10720393410010122/166375"
    ]
  ],
  "submitter": "jesper-petersen",
  "history": [],
  "commentary": "Found by a conductor-oriented search in the standard Z/2Z x Z/4Z family\r\ny^2 = x(x+u^2)(x+v^2), using the square-sum condition u+v=w^2.\r\nHere (u,v)=(3431,7178), with u+v=10609=103^2. The relevant\r\nfactorization is\r\nu=4773, v=23797, v-u=31249, v+u=103^2.\r\nPARI/GP 2.17.4 gives exact rank 5 and torsion Z/2Z x Z/4Z.\r\nConductor N=38019384450552. ",
  "created_at": "2026-09-27 07:20:35",
  "updated_at": "2026-09-27 07:20:35"
}