{
  "id": 5868,
  "curve_key": "0:2084993139251846023200",
  "ainvs": [
    "0",
    "0",
    "0",
    "0",
    "-2413186503763710675"
  ],
  "rank_lower_bound": 6,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell",
    "cubesum"
  ],
  "naive_height": 98.17810503518723,
  "faltings_height": 6.239164662057232,
  "conductor": "781872427219442258700",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "13",
    "17",
    "83",
    "127"
  ],
  "discriminant": "-2515738652041242929384430595487068830000",
  "regulator": "11646849.789127882394709675606628440880700035108843",
  "points": [
    [
      "911997529/324",
      "26008970036867/5832"
    ],
    [
      "10374125371/3249",
      "1016722847052856/185193"
    ],
    [
      "97279711/9",
      "-958557238066/27"
    ],
    [
      "200699041/144",
      "-937252456561/1728"
    ],
    [
      "5613958879/2025",
      "396098463017258/91125"
    ],
    [
      "360228619/81",
      "6742586838772/729"
    ]
  ],
  "submitter": "jesper-petersen",
  "history": [],
  "commentary": "Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, on the plane b=-1, c=-a-d. Specialization (a,b,c,d)=(20,-1,-18,-2), t=-76/129, w=16676660/16641 on w²=D_{a,d}(t), where k=(a³+1)(t-a)w/8 up to rational cubic scaling, gives the cubesum curve u³+v³=16143857730. PARI/GP 2.17.4 ellrank bounds [6,6].",
  "created_at": "2026-10-25 17:26:13",
  "updated_at": "2026-10-25 17:26:13"
}