{
  "id": 2631,
  "curve_key": "7862232961969:-21715879048183435753",
  "ainvs": [
    "1",
    "0",
    "0",
    "-163796520041",
    "25134105619020825"
  ],
  "rank_lower_bound": 2,
  "torsion": [
    6,
    2
  ],
  "naive_height": 89.07927532163033,
  "faltings_height": 5.306201044195661,
  "conductor": "33090330",
  "bad_primes": [
    "2",
    "3",
    "5",
    "7",
    "13",
    "17",
    "23",
    "31"
  ],
  "discriminant": "8346195162194017939784779335566400",
  "regulator": "9.77138801053212113687862165050606914165457",
  "points": [
    [
      "-464492",
      "31868851"
    ],
    [
      "-396242",
      "167003851"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "Exact computational search in the full 2-torsion/3-torsion family with p=-31, q=18; A=(3q-p)(p+q)^3=-186745, B=-(p+3q)(p-q)^3=2705927. The listed points are independent modulo torsion by the exact 2-descent Kummer squareclass map. The rank is a proven lower bound; no upper bound is claimed.",
  "created_at": "2026-09-26 20:13:59",
  "updated_at": "2026-09-26 20:13:59"
}