{
  "id": 6351,
  "curve_key": "590376694543561:-14344585138311261442309",
  "ainvs": [
    "1",
    "0",
    "1",
    "-12299514469658",
    "16602528070308124568"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 102.03534574327233,
  "faltings_height": 6.139304435987422,
  "conductor": "32188214579430",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "23",
    "37",
    "79",
    "83",
    "2113"
  ],
  "discriminant": "3188967518535680397712318647095434500",
  "regulator": "11422.4542415980019144401316842739924037325539408",
  "points": [
    [
      "2015127",
      "17588377"
    ],
    [
      "2860244",
      "2194603035"
    ],
    [
      "30739004",
      "169346355075"
    ],
    [
      "6738008719/2916",
      "113328880604875/157464"
    ],
    [
      "3361289456529/1666681",
      "25945564469914465/2151685171"
    ]
  ],
  "submitter": "The Wren",
  "history": [],
  "commentary": "Z/6-torsion Kubert–Tate family curve from a rational-parameter search of the family y^2+(1-c)xy-by=x^3-bx^2 with c=a/d, b=c(c+1), following the construction of David Renshaw on curve #1192 and the t=1/6916 example of Lewis Combes on curve #5970. The search covered coprime pairs (a,d) with log|Δ_min| ≤ 88.12, ranked by discriminant. Rank ≥ 5 certified: 5 independent rational points, checked exactly (independent mod 5 and mod 7); PARI ellrank bounds 5 5. The rational torsion subgroup is Z/6Z.",
  "created_at": "2026-11-01 05:05:08",
  "updated_at": "2026-11-01 05:05:08"
}