{
  "id": 4940,
  "curve_key": "-15744:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "328",
    "0"
  ],
  "rank_lower_bound": 3,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 28.992643857876104,
  "faltings_height": 0.4842940664644991,
  "conductor": "430336",
  "bad_primes": [
    "2",
    "41"
  ],
  "discriminant": "-2258403328",
  "regulator": "81.663136238143099037138914341957453614256691",
  "points": [
    [
      "9/4",
      "219/8"
    ],
    [
      "81",
      "747"
    ],
    [
      "1089/196",
      "122529/2744"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 2-isogeny from curve #13, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/13.\r\n\r\nThe source commentary attributes the original curve y^2 = x^3 - 82*x to Billing (1938), via Dujella's elliptic-curve rank-record tables.\r\n\r\nApplied the standard degree-2 isogeny with kernel generated by (0,0), sending y^2 = x^3 + A*x to y^2 = x^3 - 4*A*x. Here A = -82, giving y^2 = x^3 + 328*x. The coordinate map is X = (x^2-82)/x, Y = y*(x^2+82)/x^2.\r\n\r\nThree supplied rational points transported through this map and checked exactly on the target curve. Rational torsion C2. Same rank and conductor as the source; conductor 430336. Source-attributed existing curve.",
  "created_at": "2026-10-20 23:23:25",
  "updated_at": "2026-10-20 23:23:25"
}