{
  "id": 5697,
  "curve_key": "-14165639944143844224:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "295117498836330088",
    "0"
  ],
  "rank_lower_bound": 11,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 132.29205295199296,
  "faltings_height": 9.092578157640904,
  "conductor": "348377352477645163379230656376350976",
  "bad_primes": [
    "2",
    "53",
    "89",
    "509",
    "821",
    "1669",
    "11213"
  ],
  "discriminant": "-1644996046630803254449852383942604778024150734031454208",
  "regulator": "75268550779392375.4247424719792996503434780820855254080010778",
  "points": [
    [
      "7564034968718945870643777111049/4040456108099361045489",
      "21661394639639466851367461408544810760630859201/256829904340086167105774341922487"
    ],
    [
      "10935121418163583790132209249/35085079573662062025",
      "2297843340953999037274578509490072655149443/207818257367138406667780858875"
    ],
    [
      "90064903921040116201/457652250000",
      "2509498983500068982171940589301/309601747125000000"
    ],
    [
      "35600169939309313209/407070796441",
      "1336439740397363234532517634391/259719716616083261"
    ],
    [
      "32455202853424133601/3748500625",
      "185259153440241295546208091651/229501950765625"
    ],
    [
      "417942153142584849/1169913616",
      "491754314417000568622381689/40015725321664"
    ],
    [
      "1368966828875616481/52664049",
      "1602080029553030758014010237/382183003593"
    ],
    [
      "4080469443597230265227560369/6978144796756558569",
      "355780205209197533308590902563762422669301/18433591764280864862605595253"
    ],
    [
      "665236093198320087921/2673640306129",
      "41204097080302693958303603223723/4371741452839793383"
    ],
    [
      "491452377586587606431089/91586852290404",
      "346287217565149186304875931816603959/876495518278099901208"
    ],
    [
      "3606030230926293275530665249/12194808617510971456",
      "452937074077925051953924363066999746190641/42585571854063876874084752896"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Rational 2-isogeny partner of Michael Rubinstein’s ICARM curve #5696:\r\nhttps://elliptic-rank.icarm.cloud/curve/5696\r\n\r\nRubinstein reports rank 11 proved, with eleven independent points and a matching 2-descent upper bound, for the quartic twist y^2 = x^3 - 73779374709082522*x.\r\n\r\nWriting A = -73779374709082522, 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 + 295117498836330088*x.\r\n\r\nThe eleven 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 348377352477645163379230656376350976, equal to the source conductor.",
  "created_at": "2026-10-24 13:42:17",
  "updated_at": "2026-10-24 13:42:17"
}