{
  "id": 2536,
  "curve_key": "2769561:571985091",
  "ainvs": [
    "1",
    "-1",
    "0",
    "-57699",
    "-647595"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "naive_height": 44.502598145547395,
  "faltings_height": 1.7758480409214572,
  "conductor": "6930",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "11"
  ],
  "discriminant": "12104550027662400",
  "regulator": "7.651180632372028931438602710891376491612",
  "points": [
    [
      "-585051/2704",
      "197051061/140608"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "Exact computational full 2- and 3-torsion family at (p,q)=(-21,1); Velu quotient by rational order-3 subgroup. The mapped rank >=1 witness is non-torsion by the original exact 2-descent Kummer certificate and finite isogeny kernel. Conductor N=6930; local exponents {'2': 1, '3': 2, '5': 1, '7': 1, '11': 1}.",
  "created_at": "2026-09-26 20:07:43",
  "updated_at": "2026-09-26 20:07:43"
}