Basic invariants
Dimension: | $2$ |
Group: | 16T60 |
Conductor: | \(201\)\(\medspace = 3 \cdot 67 \) |
Artin number field: | Galois closure of 16.0.2664210032449121601.1 |
Galois orbit size: | $4$ |
Smallest permutation container: | 16T60 |
Parity: | odd |
Projective image: | $A_4$ |
Projective field: | Galois closure of 4.0.40401.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$:
\( x^{6} + 35x^{3} + 4x^{2} + 30x + 2 \)
Roots:
$r_{ 1 }$ | $=$ |
\( 25 a^{5} + 20 a^{4} + 3 a^{3} + 18 a^{2} + 16 a + 16 + \left(34 a^{5} + 31 a^{4} + 6 a^{3} + 36 a^{2} + 22 a + 10\right)\cdot 37 + \left(26 a^{5} + 13 a^{4} + 25 a^{3} + 25 a^{2} + 18 a\right)\cdot 37^{2} + \left(a^{5} + 11 a^{4} + 35 a^{3} + 8 a^{2} + 29 a + 1\right)\cdot 37^{3} + \left(4 a^{5} + 32 a^{4} + 11 a^{3} + 34 a^{2} + 27 a + 10\right)\cdot 37^{4} + \left(4 a^{5} + 7 a^{4} + 6 a^{3} + 13 a^{2} + 12 a + 28\right)\cdot 37^{5} + \left(24 a^{5} + 28 a^{4} + 27 a^{3} + 35 a^{2} + 23 a + 10\right)\cdot 37^{6} + \left(4 a^{5} + 26 a^{4} + 26 a^{3} + 13 a^{2} + 14 a + 26\right)\cdot 37^{7} + \left(4 a^{5} + 18 a^{4} + 3 a^{3} + 12 a^{2} + 9 a + 28\right)\cdot 37^{8} + \left(12 a^{5} + 4 a^{4} + 7 a^{3} + 23 a^{2} + 3 a + 8\right)\cdot 37^{9} +O(37^{10})\)
$r_{ 2 }$ |
$=$ |
\( 4 a^{5} + 33 a^{4} + 4 a^{3} + 3 a^{2} + 3 a + 28 + \left(4 a^{5} + 27 a^{4} + 16 a^{3} + 2 a^{2} + 3 a + 7\right)\cdot 37 + \left(24 a^{5} + 10 a^{3} + 33 a^{2} + 6 a + 2\right)\cdot 37^{2} + \left(24 a^{5} + 19 a^{4} + 6 a^{3} + 34 a^{2} + 28 a + 6\right)\cdot 37^{3} + \left(32 a^{5} + 14 a^{4} + 22 a^{3} + 11 a^{2} + 21 a + 35\right)\cdot 37^{4} + \left(34 a^{5} + a^{4} + 15 a^{3} + 11 a^{2} + 8 a + 31\right)\cdot 37^{5} + \left(9 a^{4} + 3 a^{3} + 18 a^{2} + 32 a + 23\right)\cdot 37^{6} + \left(26 a^{5} + a^{4} + 17 a^{3} + 11 a^{2} + 24 a + 12\right)\cdot 37^{7} + \left(9 a^{5} + 10 a^{4} + 36 a^{3} + 33 a^{2} + 30 a + 27\right)\cdot 37^{8} + \left(5 a^{5} + 6 a^{4} + 19 a^{3} + 11 a^{2} + 17 a + 19\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 24 a^{5} + 17 a^{4} + 22 a^{3} + 4 a^{2} + 14 a + 11 + \left(16 a^{5} + 11 a^{4} + 4 a^{3} + 14 a^{2} + 34 a + 14\right)\cdot 37 + \left(14 a^{5} + 20 a^{4} + 9 a^{3} + 31 a^{2} + 4 a + 30\right)\cdot 37^{2} + \left(15 a^{5} + 3 a^{4} + 17 a^{3} + 12 a^{2} + 5 a + 23\right)\cdot 37^{3} + \left(4 a^{5} + 29 a^{4} + 2 a^{3} + 14 a^{2} + 25 a + 9\right)\cdot 37^{4} + \left(27 a^{5} + 25 a^{4} + 7 a^{3} + 34 a^{2} + 25 a\right)\cdot 37^{5} + \left(15 a^{5} + 19 a^{4} + 16 a^{3} + 9 a^{2} + 28 a + 35\right)\cdot 37^{6} + \left(28 a^{5} + 23 a^{4} + 32 a^{3} + 14 a^{2} + 19 a + 5\right)\cdot 37^{7} + \left(2 a^{5} + 22 a^{4} + 14 a^{3} + 19 a^{2} + 17 a + 3\right)\cdot 37^{8} + \left(2 a^{4} + 8 a^{3} + 17 a^{2} + 3 a + 29\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 28 a^{5} + 34 a^{4} + 28 a^{3} + 21 a^{2} + 30 a + 17 + \left(23 a^{5} + 23 a^{4} + 14 a^{3} + 19 a^{2} + 11 a + 35\right)\cdot 37 + \left(12 a^{5} + 23 a^{4} + 15 a^{3} + 21 a^{2} + 21 a + 9\right)\cdot 37^{2} + \left(20 a^{5} + 15 a^{4} + 8 a^{3} + 16 a^{2} + 2 a + 25\right)\cdot 37^{3} + \left(2 a^{5} + a^{4} + 19 a^{3} + 21 a^{2} + 14 a + 29\right)\cdot 37^{4} + \left(15 a^{5} + 2 a^{4} + 13 a^{3} + 22 a^{2} + 16 a + 11\right)\cdot 37^{5} + \left(21 a^{5} + 23 a^{4} + 12 a^{3} + 23 a^{2} + 17 a + 3\right)\cdot 37^{6} + \left(5 a^{5} + 36 a^{4} + 6 a^{3} + 31 a^{2} + 25 a + 27\right)\cdot 37^{7} + \left(9 a^{5} + 6 a^{4} + 3 a^{3} + 4 a^{2} + 23 a + 14\right)\cdot 37^{8} + \left(8 a^{5} + 3 a^{4} + 35 a^{3} + 18 a^{2} + a + 28\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 13 a^{5} + 27 a^{4} + 7 a^{3} + 19 a^{2} + 24 a + 12 + \left(33 a^{5} + 19 a^{4} + 2 a^{3} + 33 a^{2} + 27 a + 23\right)\cdot 37 + \left(31 a^{5} + 15 a^{4} + 23 a^{3} + 28 a^{2} + 9 a + 25\right)\cdot 37^{2} + \left(28 a^{5} + 14 a^{4} + 16 a^{3} + 25 a^{2} + 28 a + 8\right)\cdot 37^{3} + \left(25 a^{5} + 21 a^{4} + 10 a^{3} + 11 a^{2} + 19 a + 9\right)\cdot 37^{4} + \left(29 a^{5} + 14 a^{4} + 32 a^{3} + 36 a + 1\right)\cdot 37^{5} + \left(14 a^{5} + 6 a^{4} + 20 a^{3} + 26 a^{2} + 6 a + 15\right)\cdot 37^{6} + \left(21 a^{5} + 23 a^{4} + 6 a^{3} + 11 a^{2} + 11 a + 36\right)\cdot 37^{7} + \left(14 a^{5} + 15 a^{4} + 35 a^{3} + 15 a^{2} + 5\right)\cdot 37^{8} + \left(17 a^{5} + 3 a^{4} + 35 a^{3} + 2 a^{2} + 21 a + 15\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 28 a^{5} + 5 a^{4} + 33 a^{3} + 6 a^{2} + 11 a + 5 + \left(26 a^{5} + 30 a^{4} + 12 a^{3} + 3 a^{2} + 25 a + 17\right)\cdot 37 + \left(10 a^{5} + 12 a^{4} + 12 a^{3} + 4 a^{2} + 34 a + 15\right)\cdot 37^{2} + \left(25 a^{5} + 31 a^{4} + 18 a^{3} + 9 a^{2} + 16 a + 22\right)\cdot 37^{3} + \left(22 a^{5} + 36 a^{4} + 20 a^{3} + a^{2} + 2 a + 24\right)\cdot 37^{4} + \left(20 a^{5} + 18 a^{4} + 32 a^{3} + 17 a^{2} + 31 a + 30\right)\cdot 37^{5} + \left(8 a^{5} + 5 a^{4} + 5 a^{3} + 7 a^{2} + 11 a + 5\right)\cdot 37^{6} + \left(22 a^{5} + 2 a^{4} + 16 a^{3} + 7 a^{2} + 19 a + 1\right)\cdot 37^{7} + \left(27 a^{5} + 14 a^{4} + 14 a^{3} + 27 a^{2} + 3 a + 21\right)\cdot 37^{8} + \left(20 a^{5} + 33 a^{4} + 25 a^{3} + 15 a^{2} + 6 a + 7\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 2 a^{5} + 10 a^{4} + 14 a^{3} + 23 a^{2} + 6 a + 20 + \left(5 a^{5} + 4 a^{4} + 5 a^{3} + 7 a^{2} + 31 a + 23\right)\cdot 37 + \left(8 a^{5} + 4 a^{4} + 14 a^{3} + 32 a^{2} + a + 20\right)\cdot 37^{2} + \left(11 a^{5} + 30 a^{4} + 25 a^{3} + 14 a^{2} + 5 a + 15\right)\cdot 37^{3} + \left(28 a^{5} + 29 a^{4} + 9 a^{3} + 15 a^{2} + 17 a + 33\right)\cdot 37^{4} + \left(8 a^{5} + 29 a^{4} + 6 a^{3} + 10 a^{2} + 2 a + 10\right)\cdot 37^{5} + \left(13 a^{5} + 14 a^{4} + 7 a^{3} + 20 a^{2} + 15 a + 25\right)\cdot 37^{6} + \left(4 a^{5} + 3 a^{4} + a^{3} + 29 a^{2} + 26 a + 27\right)\cdot 37^{7} + \left(9 a^{5} + 20 a^{4} + 4 a^{3} + 35 a^{2} + 13 a + 20\right)\cdot 37^{8} + \left(32 a^{5} + 28 a^{4} + 33 a^{3} + 11 a^{2} + 13 a + 1\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 8 }$ |
$=$ |
\( a^{5} + 20 a^{4} + 21 a^{3} + 24 a^{2} + 7 a + 28 + \left(33 a^{5} + 9 a^{4} + 14 a^{3} + 12 a^{2} + 27 a + 21\right)\cdot 37 + \left(24 a^{5} + 24 a^{4} + 4 a^{3} + 28 a^{2} + 12 a\right)\cdot 37^{2} + \left(15 a^{5} + 32 a^{4} + 29 a^{3} + 36 a^{2} + 33 a + 36\right)\cdot 37^{3} + \left(34 a^{5} + 10 a^{4} + 21 a^{3} + 6 a^{2} + 29 a + 30\right)\cdot 37^{4} + \left(4 a^{5} + 10 a^{4} + 17 a^{3} + 8 a^{2} + 11 a\right)\cdot 37^{5} + \left(22 a^{5} + 15 a^{4} + 33 a^{3} + 4 a^{2} + 30 a + 11\right)\cdot 37^{6} + \left(9 a^{5} + 21 a^{4} + 33 a^{3} + 19 a^{2} + 28 a + 35\right)\cdot 37^{7} + \left(13 a^{5} + 11 a^{4} + 18 a^{3} + 22 a^{2} + 35 a + 1\right)\cdot 37^{8} + \left(5 a^{5} + 5 a^{4} + 17 a^{3} + 6 a^{2} + 14 a + 19\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 14 a^{5} + 14 a^{4} + 8 a^{3} + 18 a^{2} + a + 16 + \left(33 a^{5} + 22 a^{4} + 34 a^{2} + 30 a + 17\right)\cdot 37 + \left(36 a^{5} + 31 a^{4} + 6 a^{3} + 20 a^{2} + 32 a + 35\right)\cdot 37^{2} + \left(7 a^{5} + 23 a^{4} + 15 a^{3} + 26 a^{2} + 20 a + 16\right)\cdot 37^{3} + \left(36 a^{5} + 2 a^{4} + 23 a^{3} + 9 a^{2} + 25 a + 10\right)\cdot 37^{4} + \left(19 a^{5} + 30 a^{4} + 8 a^{3} + 7 a^{2} + 22 a + 31\right)\cdot 37^{5} + \left(30 a^{5} + 28 a^{4} + 9 a^{3} + 34 a^{2} + 36 a + 31\right)\cdot 37^{6} + \left(29 a^{5} + 29 a^{4} + 14 a^{3} + 9 a^{2} + 3 a + 25\right)\cdot 37^{7} + \left(a^{5} + 6 a^{4} + 2 a^{3} + 6 a^{2} + 35 a + 32\right)\cdot 37^{8} + \left(a^{5} + 26 a^{4} + 28 a^{3} + 22 a^{2} + 34 a + 8\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 10 }$ |
$=$ |
\( 17 a^{5} + 32 a^{3} + 15 a^{2} + 7 a + 15 + \left(15 a^{5} + 16 a^{4} + 36 a^{3} + 29 a^{2} + 36 a + 6\right)\cdot 37 + \left(31 a^{5} + 14 a^{4} + 13 a^{3} + 20 a^{2} + 29 a + 6\right)\cdot 37^{2} + \left(21 a^{5} + a^{4} + 7 a^{3} + 16 a^{2} + 35 a + 19\right)\cdot 37^{3} + \left(7 a^{5} + 13 a^{4} + 36 a^{3} + 21 a^{2} + 8 a + 7\right)\cdot 37^{4} + \left(a^{5} + 6 a^{4} + 8 a^{3} + 32 a + 6\right)\cdot 37^{5} + \left(9 a^{5} + 33 a^{4} + 21 a^{3} + 35 a^{2} + 22 a + 4\right)\cdot 37^{6} + \left(33 a^{5} + 27 a^{4} + 29 a^{3} + 2 a^{2} + a + 20\right)\cdot 37^{7} + \left(31 a^{5} + 8 a^{4} + 34 a^{3} + 32 a^{2} + 15 a + 25\right)\cdot 37^{8} + \left(13 a^{5} + a^{4} + 22 a^{3} + 2 a^{2} + 19 a + 1\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 11 }$ |
$=$ |
\( 35 a^{5} + 24 a^{4} + 25 a^{3} + 22 a^{2} + 25 a + 18 + \left(33 a^{5} + 36 a^{3} + 5 a^{2} + 11 a + 23\right)\cdot 37 + \left(22 a^{5} + 17 a^{4} + 11 a^{3} + 25 a^{2} + 32 a + 30\right)\cdot 37^{2} + \left(34 a^{5} + 35 a^{4} + 18 a^{3} + 34 a^{2} + 30 a + 28\right)\cdot 37^{3} + \left(a^{5} + 15 a^{4} + 23 a^{3} + 19 a^{2} + 28 a + 27\right)\cdot 37^{4} + \left(3 a^{5} + 35 a^{4} + 20 a^{3} + 21 a + 31\right)\cdot 37^{5} + \left(23 a^{5} + 25 a^{4} + 21 a^{3} + 29 a^{2} + 24 a + 14\right)\cdot 37^{6} + \left(13 a^{5} + 26 a^{4} + 34 a^{3} + 34 a^{2} + 3 a + 33\right)\cdot 37^{7} + \left(22 a^{5} + 36 a^{3} + 29 a^{2} + 17 a + 12\right)\cdot 37^{8} + \left(35 a^{5} + 16 a^{4} + 20 a^{3} + 28 a^{2} + a + 20\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 12 }$ |
$=$ |
\( 32 a^{5} + 3 a^{4} + 25 a^{3} + 12 a^{2} + 16 a + 25 + \left(26 a^{5} + 18 a^{4} + 6 a^{3} + 10 a^{2} + 20 a + 11\right)\cdot 37 + \left(29 a^{5} + 12 a^{4} + 26 a^{3} + 8 a^{2} + 3 a + 16\right)\cdot 37^{2} + \left(28 a^{5} + 29 a^{4} + 17 a^{3} + 31 a^{2} + 35 a + 10\right)\cdot 37^{3} + \left(32 a^{5} + 20 a^{4} + 16 a^{3} + 15 a^{2} + 26 a + 24\right)\cdot 37^{4} + \left(13 a^{5} + 16 a^{4} + 26 a^{3} + 35 a^{2} + 33 a + 6\right)\cdot 37^{5} + \left(20 a^{5} + 2 a^{4} + 22 a^{3} + 32 a^{2} + 9 a + 11\right)\cdot 37^{6} + \left(4 a^{5} + 27 a^{4} + 16 a^{3} + 35 a^{2} + 24 a + 26\right)\cdot 37^{7} + \left(36 a^{5} + 26 a^{4} + 10 a^{3} + 26 a^{2} + 20 a + 5\right)\cdot 37^{8} + \left(13 a^{5} + 23 a^{4} + 22 a^{3} + 20 a^{2} + 29 a + 33\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 13 }$ |
$=$ |
\( 18 a^{5} + 10 a^{4} + 21 a^{3} + 11 a^{2} + a + 16 + \left(7 a^{5} + 35 a^{4} + 14 a^{3} + 26 a^{2} + 35 a + 29\right)\cdot 37 + \left(23 a^{5} + 5 a^{4} + 5 a^{3} + 8 a^{2} + 27 a + 3\right)\cdot 37^{2} + \left(22 a^{5} + 8 a^{3} + 23 a^{2} + 19 a + 24\right)\cdot 37^{3} + \left(a^{5} + 17 a^{4} + 16 a^{3} + 31 a^{2} + 6 a + 24\right)\cdot 37^{4} + \left(15 a^{5} + 27 a^{4} + 26 a^{3} + 26 a^{2} + 23 a + 35\right)\cdot 37^{5} + \left(10 a^{5} + 16 a^{4} + 27 a^{3} + 29 a^{2} + 32 a + 33\right)\cdot 37^{6} + \left(25 a^{5} + 8 a^{4} + 27 a^{3} + 35 a^{2} + 12 a + 20\right)\cdot 37^{7} + \left(29 a^{5} + 16 a^{4} + 26 a^{3} + 17 a^{2} + 30 a + 9\right)\cdot 37^{8} + \left(a^{5} + 7 a^{4} + 2 a^{3} + 10 a^{2} + 10 a + 4\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 14 }$ |
$=$ |
\( 9 a^{5} + 32 a^{4} + 4 a^{3} + 31 a^{2} + 26 a + 1 + \left(10 a^{5} + 6 a^{4} + 24 a^{3} + 33 a^{2} + 11 a + 16\right)\cdot 37 + \left(26 a^{5} + 24 a^{4} + 24 a^{3} + 32 a^{2} + 2 a + 2\right)\cdot 37^{2} + \left(11 a^{5} + 5 a^{4} + 18 a^{3} + 27 a^{2} + 20 a + 21\right)\cdot 37^{3} + \left(14 a^{5} + 16 a^{3} + 35 a^{2} + 34 a + 11\right)\cdot 37^{4} + \left(16 a^{5} + 18 a^{4} + 4 a^{3} + 19 a^{2} + 5 a + 16\right)\cdot 37^{5} + \left(28 a^{5} + 31 a^{4} + 31 a^{3} + 29 a^{2} + 25 a + 33\right)\cdot 37^{6} + \left(14 a^{5} + 34 a^{4} + 20 a^{3} + 29 a^{2} + 17 a + 26\right)\cdot 37^{7} + \left(9 a^{5} + 22 a^{4} + 22 a^{3} + 9 a^{2} + 33 a + 21\right)\cdot 37^{8} + \left(16 a^{5} + 3 a^{4} + 11 a^{3} + 21 a^{2} + 30 a + 12\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 15 }$ |
$=$ |
\( 10 a^{5} + 30 a^{4} + 33 a^{3} + 19 a^{2} + 5 a + 20 + \left(24 a^{5} + 10 a^{4} + 3 a^{3} + 2 a^{2} + 32 a + 20\right)\cdot 37 + \left(33 a^{5} + 25 a^{4} + 24 a^{3} + 2 a^{2} + 32 a + 7\right)\cdot 37^{2} + \left(3 a^{5} + 8 a^{3} + 13 a^{2} + 17 a + 4\right)\cdot 37^{3} + \left(7 a^{5} + 24 a^{4} + 30 a^{3} + 14 a^{2} + 36 a + 14\right)\cdot 37^{4} + \left(12 a^{5} + 24 a^{4} + 12 a^{3} + 21 a^{2} + 22 a + 6\right)\cdot 37^{5} + \left(a^{5} + 13 a^{4} + 32 a^{3} + a^{2} + 8 a + 17\right)\cdot 37^{6} + \left(25 a^{5} + 24 a^{4} + 8 a^{3} + 27 a^{2} + 16 a + 13\right)\cdot 37^{7} + \left(13 a^{5} + 31 a^{4} + 13 a^{3} + 24 a^{2} + 8 a + 18\right)\cdot 37^{8} + \left(6 a^{5} + 24 a^{4} + 22 a^{3} + 14 a^{2} + 28 a + 1\right)\cdot 37^{9} +O(37^{10})\)
| $r_{ 16 }$ |
$=$ |
\( 36 a^{5} + 17 a^{4} + 16 a^{3} + 13 a^{2} + 30 a + 12 + \left(3 a^{5} + 27 a^{4} + 22 a^{3} + 24 a^{2} + 9 a + 17\right)\cdot 37 + \left(12 a^{5} + 12 a^{4} + 32 a^{3} + 8 a^{2} + 24 a + 14\right)\cdot 37^{2} + \left(21 a^{5} + 4 a^{4} + 7 a^{3} + 3 a + 32\right)\cdot 37^{3} + \left(2 a^{5} + 26 a^{4} + 15 a^{3} + 30 a^{2} + 7 a + 29\right)\cdot 37^{4} + \left(32 a^{5} + 26 a^{4} + 19 a^{3} + 28 a^{2} + 25 a + 8\right)\cdot 37^{5} + \left(14 a^{5} + 21 a^{4} + 3 a^{3} + 32 a^{2} + 6 a + 19\right)\cdot 37^{6} + \left(27 a^{5} + 15 a^{4} + 3 a^{3} + 17 a^{2} + 8 a + 30\right)\cdot 37^{7} + \left(23 a^{5} + 25 a^{4} + 18 a^{3} + 14 a^{2} + a + 8\right)\cdot 37^{8} + \left(31 a^{5} + 31 a^{4} + 19 a^{3} + 30 a^{2} + 22 a + 10\right)\cdot 37^{9} +O(37^{10})\)
| |
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 | |||
$c1$ | $c2$ | $c3$ | $c4$ | |||
$1$ | $1$ | $()$ | $2$ | $2$ | $2$ | $2$ |
$1$ | $2$ | $(1,9)(2,10)(3,11)(4,12)(5,13)(6,14)(7,15)(8,16)$ | $-2$ | $-2$ | $-2$ | $-2$ |
$6$ | $2$ | $(1,14)(2,4)(3,8)(5,7)(6,9)(10,12)(11,16)(13,15)$ | $0$ | $0$ | $0$ | $0$ |
$4$ | $3$ | $(1,16,12)(3,14,13)(4,9,8)(5,11,6)$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ |
$4$ | $3$ | $(1,12,16)(3,13,14)(4,8,9)(5,6,11)$ | $-\zeta_{12}^{2} + 1$ | $\zeta_{12}^{2}$ | $\zeta_{12}^{2}$ | $-\zeta_{12}^{2} + 1$ |
$1$ | $4$ | $(1,3,9,11)(2,7,10,15)(4,5,12,13)(6,16,14,8)$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ |
$1$ | $4$ | $(1,11,9,3)(2,15,10,7)(4,13,12,5)(6,8,14,16)$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ | $-2 \zeta_{12}^{3}$ | $2 \zeta_{12}^{3}$ |
$6$ | $4$ | $(1,16,9,8)(2,13,10,5)(3,14,11,6)(4,15,12,7)$ | $0$ | $0$ | $0$ | $0$ |
$4$ | $6$ | $(1,4,16,9,12,8)(2,10)(3,5,14,11,13,6)(7,15)$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ |
$4$ | $6$ | $(1,8,12,9,16,4)(2,10)(3,6,13,11,14,5)(7,15)$ | $-\zeta_{12}^{2}$ | $\zeta_{12}^{2} - 1$ | $\zeta_{12}^{2} - 1$ | $-\zeta_{12}^{2}$ |
$4$ | $12$ | $(1,14,4,11,16,13,9,6,12,3,8,5)(2,7,10,15)$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ |
$4$ | $12$ | $(1,13,8,11,12,14,9,5,16,3,4,6)(2,7,10,15)$ | $\zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}$ |
$4$ | $12$ | $(1,6,4,3,16,5,9,14,12,11,8,13)(2,15,10,7)$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ |
$4$ | $12$ | $(1,5,8,3,12,6,9,13,16,11,4,14)(2,15,10,7)$ | $-\zeta_{12}$ | $\zeta_{12}^{3} - \zeta_{12}$ | $-\zeta_{12}^{3} + \zeta_{12}$ | $\zeta_{12}$ |