{
  "id": 5488,
  "curve_key": "19008:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-396",
    "0"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 29.557845666487115,
  "faltings_height": 0.5313942171820832,
  "conductor": "34848",
  "bad_primes": [
    "2",
    "3",
    "11"
  ],
  "discriminant": "3974344704",
  "regulator": "1.65402186404476473626042993306057742964569",
  "points": [
    [
      "-18",
      "36"
    ],
    [
      "198",
      "2772"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Found in an exact bounded sweep of quartic twists y^2 = x^3 + Ax over every integer A with 301 <= |A| <= 1500. The model y^2 = x^3 - 396x has rational torsion Z/2Z generated by [0,0].\r\n\r\nPARI/GP 2.17.4 returned rank bounds [2,2], conductor 34848 and minimal discriminant 3974344704. Exact quadratic-character certification at 9 good primes gives an F_2 matrix of rank 3 with torsion-row rank 1, proving the two submitted points independent modulo torsion.",
  "created_at": "2026-10-22 11:33:30",
  "updated_at": "2026-10-22 11:33:30"
}