{
  "id": 5698,
  "curve_key": "45326034212110656:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-944292379418972",
    "0"
  ],
  "rank_lower_bound": 10,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 115.05797390696445,
  "faltings_height": 7.656404903888528,
  "conductor": "1783376195657487548808627073568",
  "bad_primes": [
    "2",
    "19",
    "157",
    "197",
    "313",
    "769",
    "1669"
  ],
  "discriminant": "53888913638290018498380350346329050897245827072",
  "regulator": "26066860266400.95043120385123063317109474988074737487711563",
  "points": [
    [
      "3842242906896/47089",
      "6976912166751509928/10218313"
    ],
    [
      "6203646266944/3249",
      "-15449495046779514896/185193"
    ],
    [
      "134797653904/1521",
      "46420406115959816/59319"
    ],
    [
      "1418311883041/7056",
      "-1669253848825485007/592704"
    ],
    [
      "42905919168121/705600",
      "242517322588797086669/592704000"
    ],
    [
      "1212573166561/39204",
      "151786059872586353/7762392"
    ],
    [
      "51082410013100761/682463376",
      "-10527512802491478850045667/17828673234624"
    ],
    [
      "28275068994065816976/43934256025",
      "-150179233125121403372660929224/9208839734120125"
    ],
    [
      "6623998606369/76176",
      "15948266286209975729/21024576"
    ],
    [
      "1210331413089999424/25676537121",
      "-1009711001008826242592135824/4114382631731919"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 2-isogeny partner of Michael Rubinstein’s ICARM curve #5694:\r\nhttps://elliptic-rank.icarm.cloud/curve/5694\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 + 236073094854743*x.\r\n\r\nWriting A = 236073094854743, 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 - 944292379418972*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 1783376195657487548808627073568, equal to the source conductor.",
  "created_at": "2026-10-24 13:43:43",
  "updated_at": "2026-10-24 13:43:43"
}