{
  "id": 6348,
  "curve_key": "5128967567718649:-367320296457706223374093",
  "ainvs": [
    "1",
    "0",
    "1",
    "-106853490994139",
    "425139223106776471986"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 108.52104233994922,
  "faltings_height": 6.505125838862276,
  "conductor": "71020124709730",
  "bad_primes": [
    "2",
    "5",
    "7",
    "17",
    "31",
    "281",
    "883",
    "7759"
  ],
  "discriminant": "1135567536855015663138280717529656100",
  "regulator": "2946.18442564883709010143982079541059128391451251",
  "points": [
    [
      "23849367/4",
      "169428049/8"
    ],
    [
      "100644501/16",
      "87837580493/64"
    ],
    [
      "53926501/9",
      "2635665916/27"
    ],
    [
      "420926308/81",
      "2325912075737/729"
    ],
    [
      "722786883/121",
      "26226203038/1331"
    ]
  ],
  "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 05:03:04",
  "updated_at": "2026-11-01 05:03:04"
}