{
  "id": 6349,
  "curve_key": "334245762397878345:-193240816974492804585870021",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-6963453383289132",
    "223658354720600388167040"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 121.05195879861104,
  "faltings_height": 7.3503331083973515,
  "conductor": "299402608039938",
  "bad_primes": [
    "2",
    "3",
    "13",
    "19",
    "43",
    "97",
    "1993",
    "8101"
  ],
  "discriminant": "1208519806400047656684338282065834128",
  "regulator": "20863.5950681142071075324771197517681939380820203",
  "points": [
    [
      "28858119",
      "216177546786"
    ],
    [
      "48172512",
      "45816720"
    ],
    [
      "50221852",
      "24715771640"
    ],
    [
      "3902444995/81",
      "-17550142505/729"
    ],
    [
      "179269309920/3721",
      "-3822689419920/226981"
    ]
  ],
  "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 05:03:48",
  "updated_at": "2026-11-01 05:03:48"
}