{
  "id": 5356,
  "curve_key": "-322752:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "6724",
    "0"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 38.05391851630919,
  "faltings_height": 1.2394002880005897,
  "conductor": "53792",
  "bad_primes": [
    "2",
    "41"
  ],
  "discriminant": "-19456426971136",
  "regulator": "80.2499900143016103077517417536964496117700",
  "points": [
    [
      "1600/9",
      "-70480/27"
    ],
    [
      "705600/841",
      "-595528080/24389"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 2-isogeny partner of curve #5011: https://elliptic-rank.icarm.cloud/curve/5011. Starting from the congruent-number curve y^2 = x^3 - 41^2*x, apply the standard 2-isogeny y^2 = x^3 + A*x -> y^2 = x^3 - 4*A*x. The submitted points are the exact images of (-9,120) and (841,24360) under (x,y) -> (y^2/x^2, y*(A-x^2)/x^2). Source curve credited to David Renshaw, found in an exhaustive ellrank scan of squarefree n <= 20000.",
  "created_at": "2026-10-21 18:20:37",
  "updated_at": "2026-10-21 18:20:37"
}