{
  "id": 5534,
  "curve_key": "190967419149408:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-3978487898946",
    "0"
  ],
  "rank_lower_bound": 8,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 98.6493718471653,
  "faltings_height": 6.289021398905265,
  "conductor": "389471519250989804116224",
  "bad_primes": [
    "2",
    "3",
    "11",
    "17",
    "211",
    "433",
    "761"
  ],
  "discriminant": "4030269596169184422281391946917010850304",
  "regulator": "23441684804.9520573051976398253663644286211210893667171",
  "points": [
    [
      "6819621561/400",
      "559303745307309/8000"
    ],
    [
      "48680245357129/14868736",
      "269339082765871005755/57333846016"
    ],
    [
      "1440525647961/329476",
      "1538543896097260275/189119224"
    ],
    [
      "174460166305569/24700900",
      "2210530211789464942497/122763473000"
    ],
    [
      "12785955625/2704",
      "1310848415933525/140608"
    ],
    [
      "1534826281105881/63234304",
      "-59926297135966108287075/502839185408"
    ],
    [
      "17732681018361/240100",
      "74645419368957150141/117649000"
    ],
    [
      "26052153265161/12180100",
      "48013236636365349909/42508549000"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 2-isogeny partner of ICARM curve #5532:\r\nhttps://elliptic-rank.icarm.cloud/curve/5532\r\n\r\nMichael Rubinstein submitted #5532 as the proved rank-8 quartic twist\r\ny^2 = x^3 + 15913951595784*x, with eight independent listed points and a matching 2-descent upper bound.\r\n\r\nApplying the standard rational 2-isogeny\r\n(x,y) |-> (x + A/x, y*(x^2-A)/x^2)\r\ngives y^2 = x^3 - 4*A*x. Reducing by the rational fourth-power scaling\r\n(x,y) = (4*X,8*Y) gives the submitted minimal model\r\n\r\ny^2 = x^3 - 3978487898946*x.\r\n\r\nThe eight submitted points are the exact transported images of the eight witnesses on #5532. Rational isogeny preserves rank and conductor; the conductor is 389471519250989804116224. This is the opposite member of the quartic 2-isogeny pair.",
  "created_at": "2026-10-22 23:23:40",
  "updated_at": "2026-10-22 23:23:40"
}