bianchi_hat_lattices = 
[
    (
        "BianchiHat_2",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 4],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [2, 0, 0, 1],
        ]
    ),
    (
        "BianchiHat_15",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 8],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, -1, 2],
            [1, 0, 1, 0],
            [15, 0, 1, -2],
            [3, 1, 1, -1],
            [15, 15, 4, -8],
            [15, 15, 11, -7],
        ]
    ),
    (
        "BianchiHat_3",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 2],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 2],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
        ]
    ),
    (
        "BianchiHat_7",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 4],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, -1, 2],
            [1, 0, 1, 0],
            [1, 1, 1, -1],
            [7, 0, 1, -2],
        ]
    ),
    (
        "BianchiHat_5",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 10],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [5, 0, 0, 1],
            [2, 2, 1, 1],
        ]
    ),
    (
        "BianchiHat_10",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 20],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [10, 0, 0, 1],
            [4, 2, 0, 1],
            [8, 6, 3, 2],
            [30, 30, 10, 9],
            [40, 40, 20, 11],
        ]
    ),
    (
        "BianchiHat_6",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 12],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [6, 0, 0, 1],
            [2, 2, 0, 1],
        ]
    ),
    (
        "BianchiHat_1",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 2],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 1],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
        ]
    ),
    (
        "BianchiHat_17",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 34],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [17, 0, 0, 1],
            [4, 4, 1, 1],
            [8, 2, 1, 1],
            [68, 34, 17, 11],
            [17, 9, 1, 3],
            [19, 8, 0, 3],
            [85, 51, 0, 16],
            [136, 68, 17, 23],
            [204, 102, 0, 35],
        ]
    ),
    (
        "BianchiHat_21",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 42],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [21, 0, 0, 1],
            [6, 3, 0, 1],
            [10, 2, 1, 1],
            [6, 4, 2, 1],
            [42, 42, 21, 8],
            [14, 14, 3, 3],
            [63, 63, 21, 13],
        ]
    ),
    (
        "BianchiHat_13",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 26],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [13, 0, 0, 1],
            [6, 2, 1, 1],
            [4, 3, 0, 1],
            [4, 4, 2, 1],
            [52, 39, 13, 12],
            [52, 52, 13, 14],
        ]
    ),
    (
        "BianchiHat_14",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 28],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [14, 0, 0, 1],
            [7, 7, 0, 2],
            [6, 2, 0, 1],
            [4, 4, 2, 1],
            [28, 14, 7, 5],
        ]
    ),
    (
        "BianchiHat_30",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 60],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [5, 5, 0, 1],
            [30, 0, 0, 1],
            [9, 3, 0, 1],
            [6, 6, 3, 1],
            [8, 4, 2, 1],
            [14, 2, 0, 1],
            [50, 10, 5, 4],
        ]
    ),
    (
        "BianchiHat_33",
        [0 -1 0 0; -1 0 0 0; 0 0 2 0; 0 0 0 66],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, 0, -1],
            [1, 0, 1, 0],
            [6, 6, 3, 1],
            [33, 0, 0, 1],
            [11, 11, 0, 2],
            [8, 4, 1, 1],
            [16, 2, 1, 1],
            [11, 3, 1, 1],
            [99, 33, 0, 10],
            [121, 22, 0, 9],
            [90, 18, 3, 7],
            [37, 8, 0, 3],
            [264, 66, 0, 23],
        ]
    ),
    (
        "BianchiHat_11",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 6],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, -1, 2],
            [1, 0, 1, 0],
            [2, 1, 1, -1],
            [11, 0, 1, -2],
        ]
    ),
    (
        "BianchiHat_19",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 10],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, -1, 2],
            [1, 0, 1, 0],
            [2, 2, 1, -1],
            [19, 0, 1, -2],
            [4, 1, 1, -1],
        ]
    ),
    (
        "BianchiHat_39",
        [0 -1 0 0; -1 0 0 0; 0 0 2 1; 0 0 1 20],
        [
            [-1, 1, 0, 0],
            [0, 0, -1, 0],
            [0, 0, -1, 2],
            [1, 0, 1, 0],
            [3, 3, 2, -1],
            [39, 0, 1, -2],
            [12, 3, 1, -2],
            [9, 1, 1, -1],
            [26, 13, 3, -6],
            [39, 13, 10, -7],
        ]
    ),
]


@testset "Bianchi (see McLeod's thesis)" begin
    K,a = Hecke.rationals_as_number_field()
    
    for (name,matrix,known_roots) in bianchi_hat_lattices 
        @testset "$name" begin
            compare_vinberg_outputs(K,matrix,known_roots)
        end
    end

end

