{
  "id": 4905,
  "curve_key": "-139673318584704:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "2909860803848",
    "0"
  ],
  "rank_lower_bound": 9,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 97.71098211943968,
  "faltings_height": 6.2108222549281304,
  "conductor": "33869159591083714926428416",
  "bad_primes": [
    "2",
    "41",
    "1601",
    "2137",
    "2593"
  ],
  "discriminant": "-1576872639253872923028465068182229516288",
  "regulator": "35459217077.826196155076337269186530565688921088283081397",
  "points": [
    [
      "19321/4",
      "948446067/8"
    ],
    [
      "10042561/9",
      "-58136404643/27"
    ],
    [
      "253350889/324",
      "-9677381812021/5832"
    ],
    [
      "104182849/196",
      "-3574480304991/2744"
    ],
    [
      "233386729/441",
      "-12032827045471/9261"
    ],
    [
      "1474483201/729",
      "-74066485714237/19683"
    ],
    [
      "46121428081/40401",
      "-17809180172101477/8120601"
    ],
    [
      "143017681/441",
      "-9157554536437/9261"
    ],
    [
      "95259883853689/508369",
      "-929786136256868625831/362467097"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Obtained by a rational 2-isogeny from curve #4760, submitted by David Renshaw: https://elliptic-rank.icarm.cloud/curve/4760.\r\n\r\nThe original curve y^2 = x^3 - 727465200962*x appears as the rank-9 example in Table 2, Section 6.1, of Julián Aguirre, Álvaro Lozano-Robledo and Juan Carlos Peral, Elliptic curves of maximal rank: https://alozano.clas.uconn.edu/wp-content/uploads/sites/490/2014/01/ALP-2-23-07.pdf.\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 = -727465200962, giving y^2 = x^3 + 2909860803848*x. The coordinate map is X = (x^2+A)/x, Y = y*(x^2-A)/x^2.\r\n\r\nNine 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 33869159591083714926428416. Source-attributed existing curve.",
  "created_at": "2026-10-20 22:39:31",
  "updated_at": "2026-10-20 22:39:31"
}