{
  "id": 5699,
  "curve_key": "84646628734088256:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-1763471431960172",
    "0"
  ],
  "rank_lower_bound": 10,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 116.93178503026705,
  "faltings_height": 7.812555830830411,
  "conductor": "6219662982679319087228388539168",
  "bad_primes": [
    "2",
    "11",
    "17",
    "41",
    "61",
    "73",
    "157",
    "233",
    "353"
  ],
  "discriminant": "350982335564005545288493674483611161274816540672",
  "regulator": "446420058627.9966453961944340900047480928207506299707396994",
  "points": [
    [
      "1234960164/25",
      "-22854649390788/125"
    ],
    [
      "15453970596/361",
      "373104540306372/6859"
    ],
    [
      "18535733316/289",
      "1907436615378348/4913"
    ],
    [
      "846926404/9",
      "22056995930356/27"
    ],
    [
      "282038436",
      "4683756214692"
    ],
    [
      "5046255369/4",
      "358272190092411/8"
    ],
    [
      "11048952996/25",
      "-1156145023480956/125"
    ],
    [
      "1659829081/36",
      "-27917949795587/216"
    ],
    [
      "47252760129/784",
      "-7368070432323231/21952"
    ],
    [
      "3411590008401/75076",
      "2407765511521079673/20570824"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 2-isogeny partner of Michael Rubinstein’s ICARM curve #5695:\r\nhttps://elliptic-rank.icarm.cloud/curve/5695\r\n\r\nRubinstein reports rank 10 proved, with ten independent points and a matching 2-descent upper bound, for the quartic twist y^2 = x^3 + 440867857990043*x.\r\n\r\nWriting A = 440867857990043, the rational 2-isogeny\r\n(x,y) -> (x + A/x, y*(x^2-A)/x^2)\r\ngives the global minimal model y^2 = x^3 - 1763471431960172*x.\r\n\r\nThe ten submitted points are the exact transported images of Rubinstein’s witnesses; each was checked on the destination curve. Rational isogeny preserves rank and independence modulo torsion. The destination has rational torsion Z/2Z and conductor 6219662982679319087228388539168, equal to the source conductor.",
  "created_at": "2026-10-24 13:45:47",
  "updated_at": "2026-10-24 13:45:47"
}