{
  "id": 5313,
  "curve_key": "0:99797248800",
  "ainvs": [
    "0",
    "0",
    "0",
    "0",
    "-115506075"
  ],
  "rank_lower_bound": 4,
  "torsion": [],
  "family": "mordell",
  "families": [
    "mordell",
    "cubesum"
  ],
  "naive_height": 50.65281290549916,
  "faltings_height": 2.278723651249892,
  "conductor": "37423968300",
  "bad_primes": [
    "2",
    "3",
    "5",
    "17",
    "73"
  ],
  "discriminant": "-5763594252343230000",
  "regulator": "1055.594844700055855711794038938193356584524050",
  "points": [
    [
      "511",
      "4234"
    ],
    [
      "20659",
      "2969348"
    ],
    [
      "7081/4",
      "589621/8"
    ],
    [
      "178339/289",
      "53702888/4913"
    ]
  ],
  "submitter": "David Renshaw",
  "history": [],
  "commentary": "Cube-sum curve x^3 + y^3 = 111690 (k = 2*3^2*5*17*73; Weierstrass model y^2 = x^3 - 432k^2, minimal model shown). Found by an exhaustive scan of cube-free k <= 7*10^5: minimal-model naive height computed for every k, a Mestre-Nagao a_p sieve over primes p = 1 mod 3 to rank candidates, then PARI ellrank 2-descent. k divisible by 9 gains a factor 3^6 in the minimal model, so k = 111690 has smaller naive height than the classical smallest rank-4 k = 21691; it is the minimum naive height over all rank-4 k in the scanned range (any larger k has naive height > 57).",
  "created_at": "2026-10-21 13:03:32",
  "updated_at": "2026-10-21 13:03:32"
}