Properties

Label 2.693.16t60.a.c
Dimension $2$
Group $\SL(2,3):C_2$
Conductor $693$
Root number not computed
Indicator $0$

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Basic invariants

Dimension: $2$
Group: 16T60
Conductor: \(693\)\(\medspace = 3^{2} \cdot 7 \cdot 11 \)
Artin stem field: Galois closure of 16.0.53194395371827682204001.1
Galois orbit size: $4$
Smallest permutation container: 16T60
Parity: odd
Determinant: 1.693.6t1.c.a
Projective image: $A_4$
Projective stem field: Galois closure of 4.0.480249.1

Defining polynomial

$f(x)$$=$ \( x^{16} - 5 x^{15} + 48 x^{13} - 71 x^{12} - 144 x^{11} + 407 x^{10} + 127 x^{9} - 1131 x^{8} + \cdots + 517 \) Copy content Toggle raw display .

The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{6} + 19x^{3} + 16x^{2} + 8x + 3 \) Copy content Toggle raw display

Roots:
$r_{ 1 }$ $=$ \( 4 a^{5} + 28 a^{4} + 11 a^{3} + 15 a^{2} + 14 a + 14 + \left(25 a^{5} + 3 a^{4} + 30 a^{3} + 28 a^{2} + 4 a + 15\right)\cdot 31 + \left(30 a^{5} + 17 a^{4} + 8 a^{3} + 23 a^{2} + 28 a + 15\right)\cdot 31^{2} + \left(11 a^{4} + 25 a^{3} + 12 a^{2} + 26 a\right)\cdot 31^{3} + \left(26 a^{5} + 27 a^{4} + 17 a^{3} + 6 a^{2} + 17 a + 21\right)\cdot 31^{4} + \left(19 a^{5} + 24 a^{4} + 14 a^{3} + 23 a^{2} + 4 a + 27\right)\cdot 31^{5} + \left(24 a^{5} + 13 a^{4} + 17 a^{3} + 23 a^{2} + 20 a + 27\right)\cdot 31^{6} + \left(2 a^{5} + 9 a^{4} + 11 a^{3} + 6 a^{2} + 27 a + 2\right)\cdot 31^{7} + \left(9 a^{5} + 9 a^{4} + 2 a^{3} + 15 a^{2} + 18 a + 13\right)\cdot 31^{8} + \left(27 a^{5} + 20 a^{4} + 11 a^{3} + 6 a^{2} + 6 a + 1\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 2 }$ $=$ \( 23 a^{5} + 28 a^{4} + 14 a^{3} + 11 a^{2} + 22 a + 8 + \left(6 a^{5} + 28 a^{4} + 9 a^{3} + 17 a^{2} + 25 a + 27\right)\cdot 31 + \left(29 a^{5} + 16 a^{4} + 26 a^{3} + 11 a^{2} + 16 a + 19\right)\cdot 31^{2} + \left(20 a^{5} + 14 a^{4} + 26 a^{3} + 8 a^{2} + 30 a + 13\right)\cdot 31^{3} + \left(28 a^{5} + 10 a^{4} + 13 a^{3} + 5 a^{2} + 19 a + 13\right)\cdot 31^{4} + \left(8 a^{5} + 24 a^{4} + 26 a^{3} + 2 a^{2} + 20 a + 5\right)\cdot 31^{5} + \left(2 a^{5} + 30 a^{4} + 9 a^{3} + 3 a^{2} + 25 a + 21\right)\cdot 31^{6} + \left(18 a^{5} + 7 a^{4} + 7 a^{3} + 2 a^{2} + 14 a + 26\right)\cdot 31^{7} + \left(22 a^{5} + 10 a^{4} + 15 a^{3} + 28 a^{2} + 22 a + 27\right)\cdot 31^{8} + \left(17 a^{5} + 15 a^{4} + 3 a^{3} + a^{2} + 21 a + 11\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 3 }$ $=$ \( 3 a^{5} + 13 a^{4} + 15 a^{3} + 11 a^{2} + 2 a + 29 + \left(14 a^{5} + 28 a^{4} + 28 a^{3} + 2 a^{2} + 29 a + 3\right)\cdot 31 + \left(20 a^{5} + 27 a^{4} + 17 a^{3} + 11 a^{2} + 22 a + 4\right)\cdot 31^{2} + \left(12 a^{5} + 11 a^{4} + 3 a^{3} + 22 a^{2} + 22 a + 19\right)\cdot 31^{3} + \left(26 a^{5} + 21 a^{4} + 8 a^{2} + 4 a + 30\right)\cdot 31^{4} + \left(a^{5} + a^{4} + 19 a^{3} + 21 a^{2} + 13 a + 17\right)\cdot 31^{5} + \left(25 a^{5} + 18 a^{4} + 12 a^{3} + 10 a^{2} + 4 a + 6\right)\cdot 31^{6} + \left(8 a^{5} + 12 a^{4} + 28 a^{3} + 28 a^{2} + 12 a + 28\right)\cdot 31^{7} + \left(18 a^{5} + 15 a^{4} + 30 a^{3} + 12 a^{2} + 12 a + 11\right)\cdot 31^{8} + \left(28 a^{5} + 11 a^{4} + 26 a^{3} + 20 a^{2} + 20 a + 8\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 4 }$ $=$ \( 8 a^{5} + 16 a^{4} + 6 a^{3} + 14 a^{2} + 29 a + \left(23 a^{5} + a^{4} + a^{2} + 29 a + 27\right)\cdot 31 + \left(27 a^{4} + 5 a^{3} + 2 a^{2} + 27 a + 8\right)\cdot 31^{2} + \left(2 a^{5} + 26 a^{4} + 5 a^{3} + 10 a^{2} + 6 a + 11\right)\cdot 31^{3} + \left(2 a^{5} + 3 a^{4} + 2 a^{3} + 7 a^{2} + 14\right)\cdot 31^{4} + \left(16 a^{5} + 7 a^{4} + 17 a^{3} + 23 a^{2} + 9 a + 12\right)\cdot 31^{5} + \left(30 a^{5} + 26 a^{4} + 6 a^{3} + 26 a^{2} + 25 a + 7\right)\cdot 31^{6} + \left(9 a^{5} + 10 a^{4} + 2 a^{3} + 8 a^{2} + 28 a + 22\right)\cdot 31^{7} + \left(23 a^{5} + 16 a^{4} + 5 a^{3} + 10 a^{2} + 16 a + 13\right)\cdot 31^{8} + \left(25 a^{5} + 26 a^{4} + 28 a^{3} + 10 a^{2} + a + 23\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 5 }$ $=$ \( 18 a^{5} + 18 a^{4} + 15 a^{3} + 22 a^{2} + 9 a + 18 + \left(14 a^{5} + a^{4} + 11 a^{3} + 10 a^{2} + 7 a + 4\right)\cdot 31 + \left(25 a^{5} + 3 a^{4} + 23 a^{3} + 30 a^{2} + 18\right)\cdot 31^{2} + \left(3 a^{5} + 5 a^{4} + 19 a^{3} + 20 a^{2} + 17 a + 11\right)\cdot 31^{3} + \left(8 a^{5} + 7 a^{4} + 28 a^{3} + 18 a^{2} + 15 a + 21\right)\cdot 31^{4} + \left(23 a^{5} + 4 a^{4} + 3 a^{3} + 9 a^{2} + 19 a + 28\right)\cdot 31^{5} + \left(13 a^{5} + 14 a^{4} + 26 a^{3} + 17 a^{2} + 24 a + 4\right)\cdot 31^{6} + \left(16 a^{5} + 26 a^{4} + 9 a^{3} + 21 a^{2} + 26 a + 16\right)\cdot 31^{7} + \left(15 a^{5} + 22 a^{4} + 3 a^{3} + 23 a^{2} + 14 a + 23\right)\cdot 31^{8} + \left(28 a^{5} + 18 a^{4} + 13 a^{3} + 12 a^{2} + 24 a + 1\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 6 }$ $=$ \( a^{5} + 24 a^{4} + 16 a^{3} + 16 a^{2} + 5 a + 3 + \left(16 a^{5} + 4 a^{4} + 22 a^{3} + 7 a^{2} + 10 a + 23\right)\cdot 31 + \left(27 a^{5} + 7 a^{4} + 4 a^{3} + 29 a^{2} + 20 a + 24\right)\cdot 31^{2} + \left(29 a^{5} + 10 a^{4} + 10 a^{3} + 5 a^{2} + 17 a + 28\right)\cdot 31^{3} + \left(16 a^{5} + 9 a^{3} + 25 a^{2} + 25 a + 26\right)\cdot 31^{4} + \left(22 a^{5} + 14 a^{4} + 11 a^{3} + 22 a^{2} + 11 a + 17\right)\cdot 31^{5} + \left(16 a^{5} + a^{4} + 21 a^{3} + 11 a^{2} + 25 a + 27\right)\cdot 31^{6} + \left(4 a^{5} + 21 a^{4} + 28 a^{3} + 12 a^{2} + 18 a + 4\right)\cdot 31^{7} + \left(16 a^{5} + 23 a^{4} + 7 a^{3} + 16 a^{2} + 14 a + 12\right)\cdot 31^{8} + \left(29 a^{5} + 4 a^{4} + 28 a^{3} + 22 a^{2} + 10 a + 6\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 7 }$ $=$ \( 8 a^{5} + 6 a^{4} + 26 a^{3} + 2 a^{2} + 5 a + 26 + \left(24 a^{5} + 27 a^{4} + 29 a^{3} + 3 a^{2} + 19 a + 25\right)\cdot 31 + \left(18 a^{5} + 26 a^{4} + 9 a^{3} + 29 a^{2} + 5 a + 22\right)\cdot 31^{2} + \left(28 a^{4} + 25 a^{3} + 28 a^{2} + 24 a + 22\right)\cdot 31^{3} + \left(16 a^{5} + 3 a^{4} + 11 a^{3} + 13 a^{2} + 30 a + 3\right)\cdot 31^{4} + \left(7 a^{5} + a^{4} + 12 a^{3} + 2 a^{2} + 29 a + 7\right)\cdot 31^{5} + \left(28 a^{5} + 24 a^{4} + 14 a^{3} + 5 a^{2} + 13 a + 18\right)\cdot 31^{6} + \left(2 a^{5} + 20 a^{4} + 10 a^{3} + 12 a^{2} + 27 a + 5\right)\cdot 31^{7} + \left(19 a^{5} + 24 a^{4} + 26 a^{3} + 6 a^{2} + 24 a + 1\right)\cdot 31^{8} + \left(9 a^{5} + 4 a^{4} + 4 a^{3} + 6 a^{2} + 17 a\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 8 }$ $=$ \( 5 a^{5} + 10 a^{4} + 27 a^{3} + a^{2} + 22 a + 16 + \left(11 a^{5} + 21 a^{4} + 19 a^{3} + 25 a^{2} + 14 a + 18\right)\cdot 31 + \left(25 a^{5} + 7 a^{4} + 7 a^{3} + 13 a^{2} + 29 a + 27\right)\cdot 31^{2} + \left(12 a^{5} + 23 a^{4} + 14 a^{3} + 2 a^{2} + 15 a + 15\right)\cdot 31^{3} + \left(6 a^{5} + 24 a^{4} + 30 a^{3} + 20 a^{2} + 19 a + 26\right)\cdot 31^{4} + \left(27 a^{5} + 5 a^{4} + 9 a^{3} + 18 a^{2} + 16 a + 10\right)\cdot 31^{5} + \left(20 a^{5} + 16 a^{4} + 10 a^{3} + 16 a^{2} + 4 a + 26\right)\cdot 31^{6} + \left(24 a^{5} + 22 a^{4} + 3 a^{3} + 25 a^{2} + 19 a + 16\right)\cdot 31^{7} + \left(27 a^{5} + 24 a^{4} + 13 a^{3} + 19 a^{2} + 9 a + 8\right)\cdot 31^{8} + \left(15 a^{5} + 26 a^{4} + 30 a^{3} + 6 a^{2} + 16 a + 8\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 9 }$ $=$ \( 25 a^{5} + 29 a^{4} + 9 a^{3} + 6 a^{2} + a + 1 + \left(a^{5} + 26 a^{4} + 3 a^{3} + 12 a + 19\right)\cdot 31 + \left(7 a^{4} + 8 a^{3} + 28 a^{2} + 29 a + 18\right)\cdot 31^{2} + \left(8 a^{5} + 28 a^{4} + 23 a^{3} + 21 a^{2} + 28 a + 4\right)\cdot 31^{3} + \left(6 a^{5} + 29 a^{4} + 16 a^{3} + 29 a^{2} + 30\right)\cdot 31^{4} + \left(a^{5} + 23 a^{4} + 8 a^{2} + 2 a + 28\right)\cdot 31^{5} + \left(13 a^{5} + 27 a^{4} + 16 a^{3} + 16 a^{2} + 24 a + 28\right)\cdot 31^{6} + \left(25 a^{5} + 8 a^{4} + 18 a^{3} + 12 a^{2} + 16 a + 27\right)\cdot 31^{7} + \left(10 a^{5} + 18 a^{4} + 14 a^{3} + 28 a^{2} + 19 a + 29\right)\cdot 31^{8} + \left(30 a^{5} + 27 a^{4} + 20 a^{3} + 6 a^{2} + 23 a + 7\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 10 }$ $=$ \( 21 a^{5} + 2 a^{2} + 9 a + 28 + \left(19 a^{5} + 8 a^{4} + 16 a^{3} + 17 a^{2} + 23 a + 19\right)\cdot 31 + \left(8 a^{5} + 22 a^{3} + 24 a^{2} + 7 a + 4\right)\cdot 31^{2} + \left(5 a^{5} + 25 a^{4} + 7 a^{3} + 28 a^{2} + 27 a + 25\right)\cdot 31^{3} + \left(7 a^{5} + 13 a^{4} + 3 a^{3} + 18 a^{2} + 3 a + 21\right)\cdot 31^{4} + \left(7 a^{5} + 22 a^{3} + 15 a^{2} + 5 a + 28\right)\cdot 31^{5} + \left(30 a^{5} + 7 a^{4} + 6 a^{3} + 13 a^{2} + 11\right)\cdot 31^{6} + \left(22 a^{5} + 2 a^{4} + 24 a^{3} + 8 a^{2} + 6 a + 3\right)\cdot 31^{7} + \left(24 a^{5} + 11 a^{4} + 21 a^{3} + 27 a^{2} + 19 a + 20\right)\cdot 31^{8} + \left(12 a^{5} + 20 a^{4} + 22 a^{3} + 4 a^{2} + 2 a + 13\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 11 }$ $=$ \( 15 a^{5} + 19 a^{4} + 4 a^{3} + 8 a^{2} + 22 + \left(22 a^{5} + 15 a^{4} + 25 a^{3} + 3 a^{2} + 4 a + 25\right)\cdot 31 + \left(26 a^{5} + 25 a^{4} + 22 a^{3} + 12 a^{2} + 21 a + 9\right)\cdot 31^{2} + \left(18 a^{5} + 16 a^{4} + 10 a^{3} + 29 a^{2} + 4 a + 5\right)\cdot 31^{3} + \left(10 a^{5} + 7 a^{4} + 22 a^{3} + 23 a^{2} + 9 a + 30\right)\cdot 31^{4} + \left(26 a^{5} + 11 a^{4} + 17 a^{3} + 30 a^{2} + 8 a + 22\right)\cdot 31^{5} + \left(25 a^{5} + 16 a^{4} + 23 a^{3} + 30 a^{2} + 21 a + 30\right)\cdot 31^{6} + \left(24 a^{5} + 5 a^{4} + 19 a^{3} + 18 a^{2} + 10 a + 18\right)\cdot 31^{7} + \left(16 a^{5} + 7 a^{4} + 19 a^{3} + 12 a^{2} + 2 a + 8\right)\cdot 31^{8} + \left(14 a^{5} + 4 a^{4} + 4 a^{3} + 16 a^{2} + 17 a + 12\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 12 }$ $=$ \( 23 a^{5} + 15 a^{4} + 25 a^{3} + 17 a^{2} + 2 a + 27 + \left(7 a^{5} + 29 a^{4} + 30 a^{3} + 29 a^{2} + a + 21\right)\cdot 31 + \left(30 a^{5} + 3 a^{4} + 25 a^{3} + 28 a^{2} + 3 a + 8\right)\cdot 31^{2} + \left(28 a^{5} + 4 a^{4} + 25 a^{3} + 20 a^{2} + 24 a + 5\right)\cdot 31^{3} + \left(28 a^{5} + 27 a^{4} + 28 a^{3} + 23 a^{2} + 30 a + 18\right)\cdot 31^{4} + \left(14 a^{5} + 23 a^{4} + 13 a^{3} + 7 a^{2} + 21 a + 24\right)\cdot 31^{5} + \left(4 a^{4} + 24 a^{3} + 4 a^{2} + 5 a + 11\right)\cdot 31^{6} + \left(21 a^{5} + 20 a^{4} + 28 a^{3} + 22 a^{2} + 2 a + 8\right)\cdot 31^{7} + \left(7 a^{5} + 14 a^{4} + 25 a^{3} + 20 a^{2} + 14 a + 22\right)\cdot 31^{8} + \left(5 a^{5} + 4 a^{4} + 2 a^{3} + 20 a^{2} + 29 a + 25\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 13 }$ $=$ \( 2 a^{5} + 4 a^{4} + 30 a^{3} + 30 a^{2} + 24 a + 13 + \left(29 a^{5} + 18 a^{4} + 18 a^{3} + 30 a^{2} + 7 a + 28\right)\cdot 31 + \left(30 a^{4} + 10 a^{2} + 6 a + 19\right)\cdot 31^{2} + \left(13 a^{5} + 24 a^{4} + 16 a^{3} + 8 a^{2} + 28 a + 1\right)\cdot 31^{3} + \left(18 a^{5} + 2 a^{4} + 28 a^{3} + 16 a^{2} + 1\right)\cdot 31^{4} + \left(22 a^{5} + 25 a^{4} + 25 a^{3} + 19 a^{2} + 8 a + 1\right)\cdot 31^{5} + \left(20 a^{5} + 27 a^{4} + 5 a^{3} + 8 a^{2} + 2 a + 20\right)\cdot 31^{6} + \left(7 a^{5} + 29 a^{4} + 6 a^{3} + 6 a^{2} + a + 21\right)\cdot 31^{7} + \left(18 a^{5} + 19 a^{4} + 28 a^{3} + 19 a^{2} + 18 a + 14\right)\cdot 31^{8} + \left(a^{5} + 23 a^{4} + 3 a^{3} + 22 a^{2} + 9\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 14 }$ $=$ \( 30 a^{5} + 9 a^{4} + 13 a^{3} + 14 a^{2} + 24 a + 13 + \left(13 a^{5} + 7 a^{4} + 22 a^{3} + 14 a^{2} + 9\right)\cdot 31 + \left(11 a^{5} + 15 a^{4} + 29 a^{3} + 4 a^{2} + 4 a + 24\right)\cdot 31^{2} + \left(5 a^{5} + 14 a^{4} + 2 a^{3} + 29 a^{2} + 21 a + 7\right)\cdot 31^{3} + \left(3 a^{5} + 8 a^{4} + 13 a^{3} + 10 a^{2} + 29 a + 16\right)\cdot 31^{4} + \left(26 a^{5} + 10 a^{4} + 27 a^{3} + 2 a + 20\right)\cdot 31^{5} + \left(23 a^{5} + 19 a^{4} + 18 a^{3} + 23 a^{2} + 16 a + 9\right)\cdot 31^{6} + \left(13 a^{5} + 12 a^{4} + 15 a^{3} + 22 a^{2} + 30 a + 7\right)\cdot 31^{7} + \left(25 a^{5} + 25 a^{4} + 28 a^{3} + 26 a^{2} + 21 a + 14\right)\cdot 31^{8} + \left(20 a^{5} + 5 a^{4} + 6 a^{3} + 26 a^{2} + 20 a + 20\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 15 }$ $=$ \( 5 a^{5} + 8 a^{4} + 2 a^{3} + 18 a^{2} + 9 a + 6 + \left(29 a^{5} + 15 a^{4} + 30 a^{3} + 19 a^{2} + 11 a + 21\right)\cdot 31 + \left(16 a^{5} + 7 a^{4} + 10 a^{3} + a^{2} + 23 a + 9\right)\cdot 31^{2} + \left(4 a^{5} + 25 a^{4} + 14 a^{3} + 29 a + 4\right)\cdot 31^{3} + \left(18 a^{5} + 21 a^{4} + 20 a^{3} + 8 a^{2} + 26 a + 22\right)\cdot 31^{4} + \left(18 a^{5} + 13 a^{4} + 4 a^{3} + 29 a^{2} + 28 a + 17\right)\cdot 31^{5} + \left(23 a^{5} + 16 a^{4} + 13 a^{3} + 21 a^{2} + 7 a + 23\right)\cdot 31^{6} + \left(6 a^{5} + 28 a^{4} + 5 a^{3} + 2 a^{2} + 24 a + 24\right)\cdot 31^{7} + \left(20 a^{5} + 28 a^{4} + 18 a^{3} + 27 a + 17\right)\cdot 31^{8} + \left(26 a^{5} + 28 a^{4} + 8 a^{3} + 7 a^{2} + 19 a + 3\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display
$r_{ 16 }$ $=$ \( 26 a^{5} + 21 a^{4} + 4 a^{3} + 30 a^{2} + 9 a + 29 + \left(19 a^{5} + 9 a^{4} + 11 a^{3} + 5 a^{2} + 16 a + 18\right)\cdot 31 + \left(5 a^{5} + 23 a^{4} + 23 a^{3} + 17 a^{2} + a + 10\right)\cdot 31^{2} + \left(18 a^{5} + 7 a^{4} + 16 a^{3} + 28 a^{2} + 15 a + 8\right)\cdot 31^{3} + \left(24 a^{5} + 6 a^{4} + 10 a^{2} + 11 a + 12\right)\cdot 31^{4} + \left(3 a^{5} + 25 a^{4} + 21 a^{3} + 12 a^{2} + 14 a + 6\right)\cdot 31^{5} + \left(10 a^{5} + 14 a^{4} + 20 a^{3} + 14 a^{2} + 26 a + 2\right)\cdot 31^{6} + \left(6 a^{5} + 8 a^{4} + 27 a^{3} + 5 a^{2} + 11 a + 12\right)\cdot 31^{7} + \left(3 a^{5} + 6 a^{4} + 17 a^{3} + 11 a^{2} + 21 a + 8\right)\cdot 31^{8} + \left(15 a^{5} + 4 a^{4} + 24 a^{2} + 14 a\right)\cdot 31^{9} +O(31^{10})\) Copy content Toggle raw display

Generators of the action on the roots $r_1, \ldots, r_{ 16 }$

Cycle notation
$(1,12,9,4)(2,6,10,14)(3,8,11,16)(5,7,13,15)$
$(1,7,9,15)(2,16,10,8)(3,6,11,14)(4,13,12,5)$
$(1,11,9,3)(2,5,10,13)(4,8,12,16)(6,7,14,15)$
$(2,6,8)(4,13,15)(5,7,12)(10,14,16)$
$(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$

Character values on conjugacy classes

SizeOrderAction on $r_1, \ldots, r_{ 16 }$ Character value
$1$$1$$()$$2$
$1$$2$$(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$$-2$
$6$$2$$(1,14)(2,4)(3,7)(5,8)(6,9)(10,12)(11,15)(13,16)$$0$
$4$$3$$(1,5,15)(2,14,3)(6,11,10)(7,9,13)$$-\zeta_{12}^{2} + 1$
$4$$3$$(1,15,5)(2,3,14)(6,10,11)(7,13,9)$$\zeta_{12}^{2}$
$1$$4$$(1,11,9,3)(2,5,10,13)(4,8,12,16)(6,7,14,15)$$2 \zeta_{12}^{3}$
$1$$4$$(1,3,9,11)(2,13,10,5)(4,16,12,8)(6,15,14,7)$$-2 \zeta_{12}^{3}$
$6$$4$$(1,7,9,15)(2,16,10,8)(3,6,11,14)(4,13,12,5)$$0$
$4$$6$$(1,7,5,9,15,13)(2,11,14,10,3,6)(4,12)(8,16)$$-\zeta_{12}^{2}$
$4$$6$$(1,13,15,9,5,7)(2,6,3,10,14,11)(4,12)(8,16)$$\zeta_{12}^{2} - 1$
$4$$12$$(1,10,7,3,5,6,9,2,15,11,13,14)(4,8,12,16)$$\zeta_{12}$
$4$$12$$(1,6,13,3,15,10,9,14,5,11,7,2)(4,8,12,16)$$\zeta_{12}^{3} - \zeta_{12}$
$4$$12$$(1,2,7,11,5,14,9,10,15,3,13,6)(4,16,12,8)$$-\zeta_{12}$
$4$$12$$(1,14,13,11,15,2,9,6,5,3,7,10)(4,16,12,8)$$-\zeta_{12}^{3} + \zeta_{12}$

The blue line marks the conjugacy class containing complex conjugation.