{
  "id": 5696,
  "curve_key": "3541409986035961056:0",
  "ainvs": [
    "0",
    "0",
    "0",
    "-73779374709082522",
    "0"
  ],
  "rank_lower_bound": 11,
  "torsion": [
    2
  ],
  "family": null,
  "families": [
    "quartic"
  ],
  "naive_height": 128.1331698686333,
  "faltings_height": 8.74600456736093,
  "conductor": "348377352477645163379230656376350976",
  "bad_primes": [
    "2",
    "53",
    "89",
    "509",
    "821",
    "1669",
    "11213"
  ],
  "discriminant": "25703063228606300850778943499103199656627355219241472",
  "regulator": "36752222060250.1833128625351461424073942764072683229531255263",
  "points": [
    [
      "-716439145019689310553/18553858441561",
      "133523587872660528307180413816303/79919242374490241059"
    ],
    [
      "-540307285610747561/3434546025",
      "559018352356363977060987863/201281569795125"
    ],
    [
      "-348728370500000/1830609",
      "6619040999088603841000/2476813977"
    ],
    [
      "-1594070093713/6889",
      "1237304431990455603/571787"
    ],
    [
      "-53129485625/6241",
      "390549487961444175/493039"
    ],
    [
      "2329168721857/4624",
      "2993581265230828383/314432"
    ],
    [
      "43701244477578/1681",
      "288879684195521441526/68921"
    ],
    [
      "715161186531226/1034289",
      "17587737122838319160542/1051871913"
    ],
    [
      "1256260476802682/2968729",
      "34142977071121437479298/5115120067"
    ],
    [
      "1962974012227519833/364886404",
      "2746739951475831013959625011/6970060089208"
    ],
    [
      "208793041813916945537/456770519104",
      "2426566744892931756601863087423/308707441795400192"
    ]
  ],
  "submitter": "Michael Rubinstein",
  "history": [],
  "commentary": "Rank 11, proved: 11 independent points listed, matching the 2-descent upper bound. Torsion Z/2. Member of the family y^2 = x^3 + a x.",
  "created_at": "2026-10-24 13:20:50",
  "updated_at": "2026-10-24 13:20:50"
}