{
  "id": 1909,
  "curve_key": "109881408:-11867005440",
  "ainvs": [
    "0",
    "0",
    "0",
    "-2289196",
    "13734960"
  ],
  "rank_lower_bound": 5,
  "torsion": [
    2,
    2
  ],
  "naive_height": 55.544736699264405,
  "faltings_height": 2.6969620217293824,
  "conductor": "3463390144",
  "bad_primes": [
    "2",
    "17",
    "47",
    "89",
    "761"
  ],
  "discriminant": "767684562531605807104",
  "regulator": "619.076295241879719210365824198152207582906685956",
  "points": [
    [
      "-1515",
      "2145"
    ],
    [
      "-1498",
      "9024"
    ],
    [
      "-1363",
      "24531"
    ],
    [
      "-1244",
      "30600"
    ],
    [
      "-840",
      "36660"
    ]
  ],
  "submitter": "Natanael Wildner Fraga",
  "history": [],
  "commentary": "Found on 2026-09-25 in a AI-assisted computational search in the family y^2 = u(u+a)(u+b), with a = 1504, b = 3026 and x = u+1510.\r\n\r\nThe five submitted points were verified to be independent modulo torsion by exact full 2-descent: the square-class matrix has rank 7, including two torsion rows of rank 2. An additional certificate using quadratic characters at good primes gives the same ranks. The torsion subgroup is exactly Z/2Z x Z/2Z, verified using good-prime reduction counts.\r\n\r\nThis proves rank >= 5. Comparison data were obtained from the ICARM Elliptic Curve Rank Leaderboard, supported by NSF Grant DMS 2425401.",
  "created_at": "2026-09-25 18:19:18",
  "updated_at": "2026-09-25 18:19:18"
}