{
  "id": 1829,
  "curve_key": "2355247029262234081:907220071953197266317918479",
  "ainvs": [
    "1",
    "0",
    "0",
    "-49067646442963210",
    "-1050023235516282187744525"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    4,
    2
  ],
  "naive_height": 126.90953187453837,
  "faltings_height": 8.64009206792836,
  "conductor": "831006184614585",
  "bad_primes": [
    "3",
    "5",
    "7",
    "23",
    "29",
    "37",
    "41",
    "43",
    "101",
    "1801"
  ],
  "discriminant": "7084462464457913730834475527760868191327646920700625",
  "regulator": "1401.32324950717627088559412553780823205141849252",
  "points": [
    [
      "-153371650",
      "1693541286065"
    ],
    [
      "-91728355",
      "1636829454665"
    ],
    [
      "2934766115/4",
      "151342604005795/8"
    ],
    [
      "-67015173385/676",
      "29620585347890965/17576"
    ],
    [
      "141682885745/64",
      "53059748904859885/512"
    ]
  ],
  "submitter": "jesper-petersen",
  "history": [],
  "commentary": "Found by searching the family\r\n\\[\r\nE_{u,v}: y^2=x(x+u^2)(x+v^2)\r\n\\]\r\nwith \\(u=31820\\) and \\(v=42021\\), motivated by the \\(C_2\\times C_4\\) constructions of Eroshkin and Dujella–Peral and by minimizing the prime support of \\(uv(u-v)(u+v)\\). Here\r\n\\[\r\nv-u=10201=101^2.\r\n\\]\r\n\r\nThe global minimal model is\r\n\\[\r\n[1,0,0,-49067646442963210,-1050023235516282187744525].\r\n\\]\r\nPARI/GP 2.17.4 gives torsion \\(\\mathbf Z/2\\mathbf Z\\times\\mathbf Z/4\\mathbf Z\\), rank bounds \\([5,5]\\), conductor\r\n\\[\r\n831006184614585,\r\n\\]\r\nand minimal discriminant\r\n\\[\r\n7084462464457913730834475527760868191327646920700625.\r\n\\]\r\nThe five submitted witness points lie on the curve and have positive canonical-height determinant, hence are independent.",
  "created_at": "2026-09-25 15:41:09",
  "updated_at": "2026-09-25 15:41:09"
}