{
  "id": 5880,
  "curve_key": "0:43427620733541513784200",
  "ainvs": [
    "0",
    "0",
    "1",
    "0",
    "-50263449923080455769"
  ],
  "rank_lower_bound": 7,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell",
    "cubesum"
  ],
  "naive_height": 104.25076522796455,
  "faltings_height": 6.745219678122008,
  "conductor": "16285357775078067669075",
  "bad_primes": [
    "3",
    "5",
    "43",
    "149",
    "421",
    "607"
  ],
  "discriminant": "-1091411020009447203572425725009073266301875",
  "regulator": "78232670725.70907971437780084937059725345737484497044",
  "points": [
    [
      "8186453329/36",
      "740700703625075/216"
    ],
    [
      "676834842721/176400",
      "-184835134934633231/74088000"
    ],
    [
      "495886750389852184201/39362649369156",
      "10902974072808488856878334366001/246959923812006192696"
    ],
    [
      "332347207944524281/29878505316",
      "188065436560478258173016489/5164619157891864"
    ],
    [
      "109903006144875157849/23814439040016",
      "805377518040089100313074001411/116214557773034240064"
    ],
    [
      "59814320208592061620801/11855432342083716",
      "11412583622115309400992363029579419/1290850937723510888262264"
    ],
    [
      "9581678214396058822640145061375561/175695290324957989711944324",
      "-937767268626760222629052913333667595116233835458993/2328842831700468855748012802799951756168"
    ]
  ],
  "submitter": "jesper-petersen",
  "history": [],
  "commentary": "Burkhardt quartic 1+a^3+b^3+c^3+d^3+3abcd=0, with b=-1 and c=-a-d. Parameters (a,d)=(141/8,-7/4) and t=d=-7/4 give w=(d+1)(a^2+ad+d^2)(d^2-d+1)=-5050737/4096 on w^2=D_{a,d}(t), where D_{a,d}(t)=L_{a,d}(t)^2-4(a^3+1)d^3(t-a)(t-d)^3, L_{a,d}(t)=(d^3+1)t^2+(3a^2d^2+ad^3-2a-3d)t-2a^2d^3+a^2+3ad+3d^2. Cubic normalization of k=(a^3+1)(t-a)w/8 gives u^3+v^3=73678033305. PARI/GP 2.17.4 ellrank bounds [7,7].",
  "created_at": "2026-10-25 19:27:41",
  "updated_at": "2026-10-25 19:27:41"
}