{
  "id": 4936,
  "curve_key": "0:-180792",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "209"
  ],
  "rank_lower_bound": 2,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell"
  ],
  "naive_height": 24.21020495634728,
  "faltings_height": 0.07517298882056873,
  "conductor": "8649",
  "bad_primes": [
    "3",
    "31"
  ],
  "discriminant": "-18915363",
  "regulator": "12.0375375425360770109843151529204971456916",
  "points": [
    [
      "-23/4",
      "31/8"
    ],
    [
      "33/16",
      "913/64"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 3-isogeny from curve #4835, submitted by Daniel Błażewicz: https://elliptic-rank.icarm.cloud/curve/4835.\r\n\r\nThe source model is y^2 + y = x^3 - 8. Its public entry supplies two rational points and no commentary identifying an earlier source.\r\n\r\nCompleted the square with Y = y + 1/2, giving Y^2 = x^3 + B with B = -31/4. Applied the standard degree-3 isogeny X = (x^3+4*B)/x^2, V = Y*(x^3-8*B)/x^3, then set y' = V - 1/2. The resulting integral model is y'^2 + y' = X^3 + 209.\r\n\r\nTwo supplied rational points transported through this map and checked exactly on the target curve. Rational torsion trivial. Same rank and conductor as the source; conductor 8649. Source-attributed existing curve.",
  "created_at": "2026-10-20 23:11:36",
  "updated_at": "2026-10-20 23:11:36"
}