Basic invariants
Dimension: | $2$ |
Group: | 16T60 |
Conductor: | \(301\)\(\medspace = 7 \cdot 43 \) |
Artin number field: | Galois closure of 16.0.67380148648514522401.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | 16T60 |
Parity: | odd |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.0.90601.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{6} + 19x^{3} + 16x^{2} + 8x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 20 a^{5} + 30 a^{4} + 21 a^{3} + 13 a^{2} + 20 a + 13 + \left(3 a^{5} + 16 a^{4} + 23 a^{3} + 10 a^{2} + 17 a + 10\right)\cdot 31 + \left(19 a^{5} + 22 a^{4} + 15 a^{3} + 19 a^{2} + 15 a + 20\right)\cdot 31^{2} + \left(16 a^{5} + 18 a^{4} + 16 a^{3} + 24 a^{2} + 13 a + 12\right)\cdot 31^{3} + \left(21 a^{5} + 28 a^{4} + 7 a^{3} + 8 a^{2} + 4 a + 21\right)\cdot 31^{4} + \left(27 a^{5} + 22 a^{3} + 25 a^{2} + 15 a + 20\right)\cdot 31^{5} + \left(23 a^{5} + 13 a^{4} + 29 a^{3} + 28 a + 7\right)\cdot 31^{6} + \left(2 a^{4} + 12 a^{3} + 9 a^{2} + 24 a\right)\cdot 31^{7} + \left(18 a^{5} + 10 a^{4} + a^{3} + 16 a^{2} + 5 a + 8\right)\cdot 31^{8} + \left(27 a^{5} + 21 a^{4} + 2 a^{3} + 7 a^{2} + 12 a + 11\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 3 a^{5} + 27 a^{4} + 19 a^{3} + 23 a^{2} + 7 a + 14 + \left(27 a^{5} + 10 a^{4} + 7 a^{3} + 4 a^{2} + 22 a + 21\right)\cdot 31 + \left(20 a^{5} + 14 a^{4} + 3 a^{3} + 16 a^{2} + 29 a + 11\right)\cdot 31^{2} + \left(12 a^{5} + 2 a^{4} + 28 a^{2} + 6 a + 8\right)\cdot 31^{3} + \left(17 a^{5} + 22 a^{4} + 14 a^{3} + 8 a^{2} + 13 a + 1\right)\cdot 31^{4} + \left(7 a^{5} + 17 a^{4} + 8 a^{3} + 19 a^{2} + 22 a + 12\right)\cdot 31^{5} + \left(30 a^{5} + 8 a^{4} + 23 a^{3} + a^{2} + 26 a + 8\right)\cdot 31^{6} + \left(28 a^{5} + 19 a^{4} + 17 a^{3} + 17 a^{2} + 17 a\right)\cdot 31^{7} + \left(16 a^{5} + 24 a^{4} + 15 a^{3} + 22 a^{2} + 22 a + 1\right)\cdot 31^{8} + \left(26 a^{5} + 25 a^{4} + 8 a^{3} + 22 a^{2} + 19 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 6 a^{5} + 25 a^{4} + 4 a^{3} + 21 a^{2} + 20 a + 15 + \left(8 a^{5} + 17 a^{4} + 10 a^{3} + 18 a^{2} + 28 a + 1\right)\cdot 31 + \left(29 a^{5} + 17 a^{4} + 28 a^{3} + 27 a^{2} + 14 a + 27\right)\cdot 31^{2} + \left(25 a^{5} + 7 a^{4} + 5 a^{3} + 21 a^{2} + 24 a + 18\right)\cdot 31^{3} + \left(17 a^{5} + 17 a^{4} + 13 a^{3} + 18 a^{2} + 21 a + 15\right)\cdot 31^{4} + \left(6 a^{5} + 22 a^{4} + 10 a^{3} + 6 a^{2} + a + 9\right)\cdot 31^{5} + \left(7 a^{5} + 9 a^{4} + 20 a^{3} + 20 a^{2} + 4 a + 21\right)\cdot 31^{6} + \left(25 a^{5} + 30 a^{4} + 8 a^{3} + 5 a^{2} + a + 12\right)\cdot 31^{7} + \left(17 a^{5} + 25 a^{4} + 13 a^{3} + 18 a^{2} + 17 a + 8\right)\cdot 31^{8} + \left(22 a^{5} + 21 a^{4} + 4 a^{3} + 12 a^{2} + 2 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 14 a^{5} + 11 a^{4} + 10 a^{3} + 7 a^{2} + 17 a + 7 + \left(12 a^{5} + 25 a^{4} + 21 a^{3} + 30 a^{2} + 23 a + 15\right)\cdot 31 + \left(23 a^{5} + 29 a^{4} + 19 a^{3} + 12 a^{2} + 11 a + 5\right)\cdot 31^{2} + \left(18 a^{5} + 5 a^{4} + 19 a^{3} + 7 a^{2} + 13 a + 22\right)\cdot 31^{3} + \left(25 a^{5} + 11 a^{4} + 10 a^{3} + 30 a^{2} + 5 a + 14\right)\cdot 31^{4} + \left(5 a^{5} + 10 a^{4} + 15 a^{3} + 29 a^{2} + 10 a + 24\right)\cdot 31^{5} + \left(28 a^{5} + 24 a^{4} + 7 a^{3} + 4 a^{2} + 19 a + 3\right)\cdot 31^{6} + \left(8 a^{5} + 12 a^{4} + 2 a^{3} + 22 a^{2} + 16 a + 2\right)\cdot 31^{7} + \left(24 a^{5} + 7 a^{4} + 11 a^{3} + 30 a^{2} + 6 a + 3\right)\cdot 31^{8} + \left(12 a^{5} + 9 a^{4} + 29 a^{3} + 29 a^{2} + 23 a + 21\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 30 a^{5} + 29 a^{4} + 7 a^{3} + 6 a^{2} + 8 a + 7 + \left(22 a^{5} + 8 a^{4} + 24 a^{3} + 11 a^{2} + 6 a + 7\right)\cdot 31 + \left(9 a^{5} + 24 a^{4} + 6 a^{3} + 22 a^{2} + 4 a + 10\right)\cdot 31^{2} + \left(18 a^{5} + 11 a^{4} + 6 a^{3} + 10 a^{2} + 17 a + 7\right)\cdot 31^{3} + \left(5 a^{5} + 11 a^{4} + 14 a^{3} + 3 a^{2} + 2 a + 28\right)\cdot 31^{4} + \left(27 a^{5} + 22 a^{4} + 25 a^{3} + 25 a^{2} + 29 a + 18\right)\cdot 31^{5} + \left(19 a^{5} + 15 a^{4} + 2 a^{3} + 27 a^{2} + 16 a + 25\right)\cdot 31^{6} + \left(15 a^{5} + 12 a^{4} + 7 a^{3} + 30 a^{2} + 19 a + 1\right)\cdot 31^{7} + \left(3 a^{5} + 19 a^{4} + 15 a^{2} + 2 a + 7\right)\cdot 31^{8} + \left(8 a^{5} + 18 a^{4} + 20 a^{3} + 5 a^{2} + 24 a + 22\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 10 a^{5} + 18 a^{3} + 8 a^{2} + 24 a + 1 + \left(24 a^{5} + 21 a^{4} + a^{3} + 24 a^{2} + 6 a + 5\right)\cdot 31 + \left(29 a^{5} + 22 a^{4} + 17 a^{3} + 8 a^{2} + 30 a + 19\right)\cdot 31^{2} + \left(28 a^{5} + 10 a^{4} + 21 a^{3} + 5 a^{2} + 12 a + 14\right)\cdot 31^{3} + \left(25 a^{5} + 24 a^{4} + 7 a^{3} + 9 a^{2} + 17 a + 10\right)\cdot 31^{4} + \left(20 a^{5} + 15 a^{4} + a^{2} + 16 a + 18\right)\cdot 31^{5} + \left(24 a^{5} + 25 a^{4} + 15 a^{3} + 12 a^{2} + 7 a + 12\right)\cdot 31^{6} + \left(16 a^{4} + 21 a^{3} + 15 a + 3\right)\cdot 31^{7} + \left(5 a^{5} + 14 a^{4} + a^{3} + 24 a^{2} + 27 a + 1\right)\cdot 31^{8} + \left(5 a^{5} + 18 a^{4} + 18 a^{3} + 29 a^{2} + 17 a + 13\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 22 a^{5} + 13 a^{4} + a^{3} + 23 a^{2} + 10 a + 20 + \left(3 a^{5} + 15 a^{4} + 27 a^{3} + 16 a^{2} + 6 a + 27\right)\cdot 31 + \left(3 a^{5} + 9 a^{4} + 15 a^{3} + 17 a^{2} + 7 a + 12\right)\cdot 31^{2} + \left(2 a^{5} + 19 a^{4} + 10 a^{3} + 30 a^{2} + 9 a + 4\right)\cdot 31^{3} + \left(22 a^{5} + 12 a^{4} + 12 a^{3} + 19 a^{2} + 8 a + 9\right)\cdot 31^{4} + \left(25 a^{5} + 27 a^{4} + 10 a^{3} + 6 a^{2} + a + 5\right)\cdot 31^{5} + \left(18 a^{5} + 20 a^{4} + 14 a^{3} + 13 a^{2} + 20 a + 14\right)\cdot 31^{6} + \left(26 a^{5} + 4 a^{4} + 27 a^{3} + 15 a^{2} + 4 a + 1\right)\cdot 31^{7} + \left(23 a^{5} + 6 a^{4} + 9 a^{3} + 17 a^{2} + 9 a + 20\right)\cdot 31^{8} + \left(4 a^{5} + 24 a^{4} + 8 a^{3} + 17 a^{2} + 11 a + 19\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 27 a^{5} + 25 a^{4} + 14 a^{3} + 17 a^{2} + 13 a + 7 + \left(23 a^{5} + 14 a^{4} + 6 a^{3} + 15 a^{2} + 25 a + 10\right)\cdot 31 + \left(10 a^{5} + 2 a^{4} + 16 a^{2} + 4 a + 21\right)\cdot 31^{2} + \left(27 a^{5} + 25 a^{4} + a^{3} + 28 a^{2} + 30 a + 4\right)\cdot 31^{3} + \left(9 a^{5} + 8 a^{4} + 15 a^{3} + 18 a^{2} + 30 a\right)\cdot 31^{4} + \left(19 a^{5} + 20 a^{4} + 29 a^{3} + 18 a^{2} + 26 a + 18\right)\cdot 31^{5} + \left(2 a^{5} + 17 a^{4} + 10 a^{3} + 2 a^{2} + 20 a + 5\right)\cdot 31^{6} + \left(18 a^{5} + 13 a^{4} + 17 a^{3} + 30 a^{2} + 15 a + 23\right)\cdot 31^{7} + \left(29 a^{5} + 8 a^{3} + 18 a^{2} + 25 a + 8\right)\cdot 31^{8} + \left(4 a^{5} + 14 a^{4} + 19 a^{3} + 11 a^{2} + 10 a + 13\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 12 a^{5} + 28 a^{4} + 22 a^{3} + 3 a^{2} + 10 a + 15 + \left(14 a^{5} + 23 a^{4} + 13 a^{3} + 19 a^{2} + 27 a + 4\right)\cdot 31 + \left(7 a^{5} + 19 a^{4} + 6 a^{3} + 21 a^{2} + 26 a\right)\cdot 31^{2} + \left(17 a^{5} + 10 a^{4} + 14 a^{3} + 18 a^{2} + 18 a + 12\right)\cdot 31^{3} + \left(20 a^{5} + 4 a^{4} + 26 a^{3} + 26 a^{2} + 22 a + 13\right)\cdot 31^{4} + \left(23 a^{4} + 3 a^{3} + 3 a^{2} + 7 a + 16\right)\cdot 31^{5} + \left(11 a^{5} + 16 a^{4} + 6 a^{3} + 10 a^{2} + 28 a + 21\right)\cdot 31^{6} + \left(15 a^{4} + 10 a^{3} + 19 a^{2} + 14 a + 2\right)\cdot 31^{7} + \left(24 a^{5} + 20 a^{4} + 15 a^{3} + 16 a^{2} + 13 a + 7\right)\cdot 31^{8} + \left(27 a^{5} + 2 a^{4} + 25 a^{3} + 27 a^{2} + 30 a + 5\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 10 }$ | $=$ | \( 18 a^{5} + 30 a^{4} + 30 a^{3} + 10 a^{2} + 17 a + 18 + \left(18 a^{5} + 13 a^{4} + 10 a^{3} + 28 a^{2} + 8 a + 29\right)\cdot 31 + \left(15 a^{5} + 11 a^{3} + 30 a + 18\right)\cdot 31^{2} + \left(25 a^{5} + 20 a^{4} + 5 a^{3} + 26 a^{2} + 5 a + 5\right)\cdot 31^{3} + \left(2 a^{5} + 21 a^{4} + 25 a^{3} + 13 a^{2} + 23 a + 26\right)\cdot 31^{4} + \left(22 a^{5} + 22 a^{4} + 9 a^{3} + 14 a^{2} + 19 a + 14\right)\cdot 31^{5} + \left(25 a^{5} + 18 a^{4} + 30 a^{3} + 23 a^{2} + 4 a + 29\right)\cdot 31^{6} + \left(9 a^{5} + 20 a^{4} + 18 a^{3} + 18 a^{2} + a + 6\right)\cdot 31^{7} + \left(23 a^{5} + 3 a^{4} + 4 a^{3} + 25 a^{2} + 5 a + 5\right)\cdot 31^{8} + \left(29 a^{5} + 19 a^{3} + 21 a^{2} + 18 a + 29\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 11 }$ | $=$ | \( 3 a^{5} + 12 a^{4} + 9 a^{2} + 22 a + 25 + \left(21 a^{5} + 14 a^{4} + 2 a^{3} + 8 a^{2} + 18 a + 4\right)\cdot 31 + \left(16 a^{5} + 26 a^{4} + 19 a^{3} + 19 a^{2} + 18 a + 17\right)\cdot 31^{2} + \left(6 a^{5} + 26 a^{4} + 24 a^{3} + 3 a^{2} + 3 a + 14\right)\cdot 31^{3} + \left(26 a^{5} + 3 a^{4} + 28 a^{3} + 28 a^{2} + 20 a + 30\right)\cdot 31^{4} + \left(26 a^{5} + 3 a^{4} + 4 a^{3} + 9 a^{2} + 15 a + 3\right)\cdot 31^{5} + \left(16 a^{5} + 5 a^{4} + 17 a^{3} + 30 a^{2} + 25 a + 6\right)\cdot 31^{6} + \left(15 a^{5} + 13 a^{4} + 7 a^{3} + 5 a^{2} + 11 a + 18\right)\cdot 31^{7} + \left(26 a^{5} + 22 a^{4} + 20 a^{3} + 9 a^{2} + 13 a + 1\right)\cdot 31^{8} + \left(20 a^{5} + 15 a^{4} + 14 a^{3} + 28 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 12 }$ | $=$ | \( 14 a^{5} + 5 a^{4} + 12 a^{3} + 5 a^{2} + 13 a + 24 + \left(29 a^{5} + 4 a^{4} + 23 a^{3} + 24 a^{2} + 20 a + 4\right)\cdot 31 + \left(28 a^{5} + 21 a^{4} + 19 a^{3} + 21 a^{2} + 12 a + 2\right)\cdot 31^{2} + \left(25 a^{5} + 20 a^{4} + 3 a^{3} + 18 a^{2} + 12 a + 13\right)\cdot 31^{3} + \left(2 a^{5} + 8 a^{4} + 13 a^{3} + 12 a^{2} + 19 a + 9\right)\cdot 31^{4} + \left(28 a^{5} + 14 a^{4} + 11 a^{3} + 20 a^{2} + 9 a + 17\right)\cdot 31^{5} + \left(13 a^{5} + 24 a^{4} + 5 a^{3} + 19 a^{2} + 18 a + 8\right)\cdot 31^{6} + \left(17 a^{5} + 9 a^{4} + 15 a^{3} + 20 a^{2} + 30 a + 5\right)\cdot 31^{7} + \left(18 a^{5} + 9 a^{4} + 25 a^{3} + 27 a^{2} + 19 a + 29\right)\cdot 31^{8} + \left(20 a^{5} + 4 a^{4} + 5 a^{3} + 10 a^{2} + 8 a + 13\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 13 }$ | $=$ | \( a^{5} + 2 a^{4} + 24 a^{3} + 25 a^{2} + 23 a + 23 + \left(8 a^{5} + 22 a^{4} + 6 a^{3} + 19 a^{2} + 24 a\right)\cdot 31 + \left(21 a^{5} + 6 a^{4} + 24 a^{3} + 8 a^{2} + 26 a + 29\right)\cdot 31^{2} + \left(12 a^{5} + 19 a^{4} + 24 a^{3} + 20 a^{2} + 13 a + 23\right)\cdot 31^{3} + \left(25 a^{5} + 19 a^{4} + 16 a^{3} + 27 a^{2} + 28 a + 19\right)\cdot 31^{4} + \left(3 a^{5} + 8 a^{4} + 5 a^{3} + 5 a^{2} + a + 28\right)\cdot 31^{5} + \left(11 a^{5} + 15 a^{4} + 28 a^{3} + 3 a^{2} + 14 a + 1\right)\cdot 31^{6} + \left(15 a^{5} + 18 a^{4} + 23 a^{3} + 11 a + 22\right)\cdot 31^{7} + \left(27 a^{5} + 11 a^{4} + 30 a^{3} + 15 a^{2} + 28 a + 27\right)\cdot 31^{8} + \left(22 a^{5} + 12 a^{4} + 10 a^{3} + 25 a^{2} + 6 a + 14\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 14 }$ | $=$ | \( 15 a^{5} + 9 a^{4} + 18 a^{3} + 12 a^{2} + 28 a + 27 + \left(28 a^{5} + 10 a^{4} + 3 a^{3} + 18 a^{2} + 25 a + 9\right)\cdot 31 + \left(26 a^{5} + 6 a^{4} + 20 a^{3} + 2 a^{2} + 4 a + 25\right)\cdot 31^{2} + \left(17 a^{5} + 21 a^{4} + 17 a^{3} + 5 a^{2} + 26 a + 30\right)\cdot 31^{3} + \left(25 a^{5} + 27 a^{4} + 19 a^{3} + 25 a^{2} + 8 a + 16\right)\cdot 31^{4} + \left(4 a^{5} + 26 a^{4} + 19 a^{3} + 24 a^{2} + 8 a + 7\right)\cdot 31^{5} + \left(2 a^{5} + 3 a^{4} + 27 a^{3} + 5 a^{2} + 18 a + 20\right)\cdot 31^{6} + \left(25 a^{5} + 10 a^{4} + 6 a^{3} + 7 a^{2} + 17 a + 1\right)\cdot 31^{7} + \left(13 a^{4} + 21 a^{3} + 14 a^{2} + 8 a + 6\right)\cdot 31^{8} + \left(11 a^{5} + 23 a^{4} + 20 a^{3} + 9 a^{2} + 12 a + 11\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 15 }$ | $=$ | \( 9 a^{5} + 18 a^{4} + 30 a^{3} + 8 a^{2} + 21 a + 9 + \left(27 a^{5} + 15 a^{4} + 3 a^{3} + 14 a^{2} + 24 a + 24\right)\cdot 31 + \left(27 a^{5} + 21 a^{4} + 15 a^{3} + 13 a^{2} + 23 a + 14\right)\cdot 31^{2} + \left(28 a^{5} + 11 a^{4} + 20 a^{3} + 21 a + 5\right)\cdot 31^{3} + \left(8 a^{5} + 18 a^{4} + 18 a^{3} + 11 a^{2} + 22 a + 17\right)\cdot 31^{4} + \left(5 a^{5} + 3 a^{4} + 20 a^{3} + 24 a^{2} + 29 a + 6\right)\cdot 31^{5} + \left(12 a^{5} + 10 a^{4} + 16 a^{3} + 17 a^{2} + 10 a + 6\right)\cdot 31^{6} + \left(4 a^{5} + 26 a^{4} + 3 a^{3} + 15 a^{2} + 26 a + 17\right)\cdot 31^{7} + \left(7 a^{5} + 24 a^{4} + 21 a^{3} + 13 a^{2} + 21 a + 23\right)\cdot 31^{8} + \left(26 a^{5} + 6 a^{4} + 22 a^{3} + 13 a^{2} + 19 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 16 }$ | $=$ | \( 13 a^{5} + 15 a^{4} + 18 a^{3} + 27 a^{2} + 26 a + \left(5 a^{5} + 12 a^{4} + 30 a^{3} + 14 a^{2} + 22 a + 9\right)\cdot 31 + \left(19 a^{5} + 2 a^{4} + 24 a^{3} + 18 a^{2} + 16 a + 12\right)\cdot 31^{2} + \left(24 a^{5} + 16 a^{4} + 24 a^{3} + 28 a^{2} + 17 a + 18\right)\cdot 31^{3} + \left(20 a^{5} + 7 a^{4} + 4 a^{3} + 15 a^{2} + 29 a + 13\right)\cdot 31^{4} + \left(15 a^{5} + 8 a^{4} + 19 a^{3} + 11 a^{2} + 25\right)\cdot 31^{5} + \left(30 a^{5} + 18 a^{4} + 23 a^{3} + 23 a^{2} + 15 a + 23\right)\cdot 31^{6} + \left(3 a^{5} + 21 a^{4} + 15 a^{3} + 29 a^{2} + 18 a + 4\right)\cdot 31^{7} + \left(12 a^{5} + 2 a^{4} + 16 a^{3} + 23 a^{2} + 20 a + 28\right)\cdot 31^{8} + \left(7 a^{5} + 29 a^{4} + 18 a^{3} + a^{2} + a + 17\right)\cdot 31^{9} +O(31^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 16 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 16 }$ | Character values | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ | $2$ | $2$ |
$1$ | $2$ | $(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$ | $-2$ | $-2$ | $-2$ | $-2$ |
$6$ | $2$ | $(1,6)(2,7)(3,16)(4,13)(5,12)(8,11)(9,14)(10,15)$ | $0$ | $0$ | $0$ | $0$ |
$4$ | $3$ | $(2,8,5)(4,14,7)(6,15,12)(10,16,13)$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ |
$4$ | $3$ | $(2,5,8)(4,7,14)(6,12,15)(10,13,16)$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $4$ | $(1,3,9,11)(2,12,10,4)(5,15,13,7)(6,16,14,8)$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ |
$1$ | $4$ | $(1,11,9,3)(2,4,10,12)(5,7,13,15)(6,8,14,16)$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ |
$6$ | $4$ | $(1,8,9,16)(2,13,10,5)(3,6,11,14)(4,15,12,7)$ | $0$ | $0$ | $0$ | $0$ |
$4$ | $6$ | $(1,9)(2,13,8,10,5,16)(3,11)(4,15,14,12,7,6)$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ |
$4$ | $6$ | $(1,9)(2,16,5,10,8,13)(3,11)(4,6,7,12,14,15)$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ |
$4$ | $12$ | $(1,3,9,11)(2,6,13,4,8,15,10,14,5,12,16,7)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$4$ | $12$ | $(1,3,9,11)(2,15,16,4,5,6,10,7,8,12,13,14)$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ |
$4$ | $12$ | $(1,11,9,3)(2,14,13,12,8,7,10,6,5,4,16,15)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$4$ | $12$ | $(1,11,9,3)(2,7,16,12,5,14,10,15,8,4,13,6)$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ |