{
  "id": 6350,
  "curve_key": "9683198520352641:-952858149725147287162689",
  "ainvs": [
    "1",
    "-1",
    "1",
    "-201733302507347",
    "1102845131133727579571"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 110.42750600183071,
  "faltings_height": 6.634010890814739,
  "conductor": "108489104061030",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11",
    "17",
    "457",
    "571",
    "3529"
  ],
  "discriminant": "1579992780436054718399732413809000000",
  "regulator": "3260.75884403738848074164779128445666047770347072",
  "points": [
    [
      "8198511",
      "4532494"
    ],
    [
      "8335531",
      "668553514"
    ],
    [
      "14832285",
      "37056482212"
    ],
    [
      "478030911",
      "10446828003394"
    ],
    [
      "1385284111/169",
      "27095461002/2197"
    ]
  ],
  "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 05:04:35",
  "updated_at": "2026-11-01 05:04:35"
}