{
  "id": 5589,
  "curve_key": "0:-19910208147266989944",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "23044222392670127"
  ],
  "rank_lower_bound": 11,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell"
  ],
  "naive_height": 88.87552849264493,
  "faltings_height": 5.463949950178707,
  "conductor": "6422206327230105208457479480347",
  "bad_primes": [
    "3",
    "7",
    "69672630060983"
  ],
  "discriminant": "-229407632214986588151309624517475187",
  "regulator": "126155696643774763.945597970601348354138215830131242159519157",
  "points": [
    [
      "-2543412696648263/9117194256",
      "31794243696073130851315/870546176339904"
    ],
    [
      "-29579061886967/320123664",
      "854464460153375656187/5727652596288"
    ],
    [
      "1013943136594705/2883045636",
      "39932956816882581189509/154802252379384"
    ],
    [
      "1838103935972078881/1502394421284",
      "2507670868403086068011254181/1841517894845067048"
    ],
    [
      "9518650483950069793/1227402525924",
      "29367961276994234606867452435/1359817165225732968"
    ],
    [
      "34007615772667965241/73234400444100",
      "219958278124045469825194453711/626718761024479011000"
    ],
    [
      "65194598957723857188457/28335885769336896",
      "16662029424037660902744315919364021/4769855138352169217170944"
    ],
    [
      "1589426003933921897015833/4070921255649417744",
      "2360085624903562004331840635757748387/8213704090523116337577198528"
    ],
    [
      "655481986898790803675827129/346754687591193299856",
      "16810501959296312474912105372143241700467/6457040511350748633090608354496"
    ],
    [
      "200404010329832031800962194025/295277858144367236334144",
      "92961564040250090181253369939128806486777419/160452438943909736099000829349513728"
    ],
    [
      "2097191413686528077434292443071262993/475943686249541178764435720004",
      "3037495105636901122386780149068296145759660081464519397/328347228548343478782038504420467742850910008"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 3-isogeny partner of jesper-petersen’s curve #5585: https://elliptic-rank.icarm.cloud/curve/5585.\r\n\r\nThe source is the specialization t = -93/13 of a generic rank-4 Mordell family obtained from the Burkhardt quartic, with affine parameters (a,b,c,d) = (-10,-32,-9,-4). Jesper-petersen describes finding it by varying the Burkhardt parameters using the rational parametrization of Bruin–Nasserden, followed by a sixth-power-residue Mestre–Nagao sieve and conductor-aware candidate selection.\r\n\r\nCompleting the square in the source model gives v^2 = x^3 + B, where B = -3413958872988167/4. Applying the rational 3-isogeny\r\nX = (x^3 + 4B)/x^2,\r\nV = v(x^3 - 8B)/x^3,\r\nand writing Y = V - 1/2 gives Y^2 + Y = X^3 + 23044222392670127.\r\n\r\nThe 11 submitted points are the exact images of the source’s independent witness points. Every image was checked on the resulting model. Independence modulo torsion is preserved by the isogeny, giving rank at least 11.",
  "created_at": "2026-10-23 21:21:53",
  "updated_at": "2026-10-23 21:21:53"
}