{
  "id": 6346,
  "curve_key": "13658853797396582881:-50480236932384704551994856529",
  "ainvs": [
    "1",
    "0",
    "0",
    "-284559454112428810",
    "58426200129509749833217572"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    6
  ],
  "family": null,
  "families": [],
  "naive_height": 132.18275884530084,
  "faltings_height": 8.068021877373525,
  "conductor": "449385702620890",
  "bad_primes": [
    "2",
    "5",
    "7",
    "19",
    "61",
    "457",
    "997",
    "12157"
  ],
  "discriminant": "206685441297902608741671598592000",
  "regulator": "161078.499092975557026531141937390174855537521985",
  "points": [
    [
      "24933858502/81",
      "3365905331530/729"
    ],
    [
      "36140716472/169",
      "5957182126492702/2197"
    ],
    [
      "517375107444/1681",
      "417169906984758/68921"
    ],
    [
      "16720053569936/54289",
      "-1931651218891174/12649337"
    ],
    [
      "4027012308703969/13075456",
      "-7045006484102488767/47280848896"
    ]
  ],
  "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 04:45:31",
  "updated_at": "2026-11-01 04:45:31"
}