{
  "id": 2540,
  "curve_key": "19773000289:2780409973223663",
  "ainvs": [
    "1",
    "0",
    "0",
    "-411937506",
    "-3218101426764"
  ],
  "rank_lower_bound": 1,
  "torsion": [
    2,
    2
  ],
  "naive_height": 71.12274966690298,
  "faltings_height": 3.427623531422753,
  "conductor": "6630",
  "bad_primes": [
    "2",
    "3",
    "5",
    "13",
    "17"
  ],
  "discriminant": "441370202660156250000",
  "regulator": "7.602028319163792105439695578195627795514",
  "points": [
    [
      "-12759446/1089",
      "200140234/35937"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "Exact computational full 2- and 3-torsion family at (p,q)=(-13,4); 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=6630; local exponents {'2': 1, '3': 1, '5': 1, '13': 1, '17': 1}.",
  "created_at": "2026-09-26 20:07:58",
  "updated_at": "2026-09-26 20:07:58"
}