# Properties

 Label 2.819.16t60.a.a Dimension $2$ Group ${group['tex_name']}$ Conductor $819$ Root number not computed Indicator $0$

# Related objects

## Basic invariants

 Dimension: $2$ Group: 16T60 Conductor: $$819$$$$\medspace = 3^{2} \cdot 7 \cdot 13$$ Artin stem field: 16.0.202428293557942478212641.1 Galois orbit size: $4$ Smallest permutation container: 16T60 Parity: odd Determinant: 1.819.6t1.f.b Projective image: A_4 Projective stem field: 4.0.670761.1

## Defining polynomial

 $f(x)$ $=$ $$x^{16} - 7 x^{15} + 31 x^{14} - 93 x^{13} + 202 x^{12} - 300 x^{11} + 277 x^{10} + 65 x^{9} - 605 x^{8} + 1100 x^{7} - 946 x^{6} + 740 x^{5} + 640 x^{4} - 1463 x^{3} + 715 x^{2} - 42 x + 28$$  .

The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.

Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: $$x^{6} + 35 x^{3} + 4 x^{2} + 30 x + 2$$

Roots:
 $r_{ 1 }$ $=$ $$17 a^{5} + a^{4} + 7 a^{3} + 22 a^{2} + 6 a + 15 + \left(33 a^{5} + 19 a^{4} + a^{3} + 5 a^{2} + 25 a + 7\right)\cdot 37 + \left(34 a^{5} + 30 a^{4} + 32 a^{3} + 20 a^{2} + 25 a + 20\right)\cdot 37^{2} + \left(3 a^{5} + 25 a^{4} + 4 a^{3} + 6 a^{2} + 9 a + 15\right)\cdot 37^{3} + \left(19 a^{5} + 33 a^{4} + 3 a^{3} + 8 a\right)\cdot 37^{4} + \left(36 a^{4} + 24 a^{3} + 24 a^{2} + 11 a + 17\right)\cdot 37^{5} + \left(34 a^{5} + 23 a^{4} + 12 a^{3} + 8 a^{2} + 3 a + 28\right)\cdot 37^{6} + \left(15 a^{5} + 23 a^{4} + 33 a^{3} + 29 a^{2} + 9 a + 3\right)\cdot 37^{7} + \left(24 a^{5} + a^{4} + 33 a^{3} + 25 a^{2} + 14 a + 33\right)\cdot 37^{8} + \left(17 a^{5} + 30 a^{4} + 28 a^{3} + 18 a^{2} + 33 a + 24\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 2 }$ $=$ $$26 a^{5} + 6 a^{4} + 2 a^{3} + 23 a^{2} + 31 a + 24 + \left(2 a^{5} + a^{4} + 9 a^{3} + 11 a^{2} + 7 a\right)\cdot 37 + \left(31 a^{5} + 6 a^{4} + 27 a^{3} + 24 a^{2} + 5 a + 22\right)\cdot 37^{2} + \left(22 a^{5} + 22 a^{4} + 34 a^{3} + 14 a^{2} + 16 a + 26\right)\cdot 37^{3} + \left(27 a^{5} + 30 a^{4} + 29 a^{3} + 3 a^{2} + 10 a + 32\right)\cdot 37^{4} + \left(16 a^{5} + a^{4} + 21 a^{3} + 26 a^{2} + 4 a + 11\right)\cdot 37^{5} + \left(34 a^{5} + 9 a^{4} + 30 a^{3} + 29 a^{2} + 25 a + 11\right)\cdot 37^{6} + \left(35 a^{5} + 6 a^{4} + a^{3} + 22 a^{2} + 28 a + 23\right)\cdot 37^{7} + \left(13 a^{5} + 34 a^{4} + 30 a^{3} + 24 a^{2} + 21 a\right)\cdot 37^{8} + \left(12 a^{5} + 22 a^{4} + 28 a^{3} + 33 a^{2} + 9 a + 20\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 3 }$ $=$ $$29 a^{4} + a^{3} + 21 a^{2} + 34 a + 16 + \left(18 a^{5} + 12 a^{4} + 11 a^{3} + 10 a^{2} + 10 a + 35\right)\cdot 37 + \left(8 a^{5} + 28 a^{4} + 13 a^{3} + 35 a^{2} + 13 a + 18\right)\cdot 37^{2} + \left(30 a^{5} + 31 a^{4} + 21 a^{3} + 21 a^{2} + 11 a + 27\right)\cdot 37^{3} + \left(22 a^{5} + 36 a^{4} + 32 a^{3} + 8 a^{2} + 29 a + 18\right)\cdot 37^{4} + \left(23 a^{5} + 5 a^{4} + 20 a^{2} + 10 a + 13\right)\cdot 37^{5} + \left(12 a^{5} + a^{4} + 10 a^{3} + 11 a^{2} + 4 a + 22\right)\cdot 37^{6} + \left(15 a^{5} + 11 a^{4} + 22 a^{3} + 4 a^{2} + 15 a + 8\right)\cdot 37^{7} + \left(22 a^{5} + 34 a^{4} + 15 a^{3} + 22 a^{2} + 31 a + 33\right)\cdot 37^{8} + \left(16 a^{5} + 22 a^{4} + 24 a^{3} + 14 a^{2} + 35 a + 11\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 4 }$ $=$ $$34 a^{5} + 33 a^{4} + 27 a^{3} + 27 a^{2} + 32 a + 17 + \left(21 a^{5} + a^{4} + 32 a^{3} + 29 a^{2} + a + 12\right)\cdot 37 + \left(36 a^{5} + 30 a^{4} + 15 a^{3} + a^{2} + 2 a + 27\right)\cdot 37^{2} + \left(19 a^{5} + 14 a^{4} + 10 a^{3} + 21 a^{2} + 15 a + 5\right)\cdot 37^{3} + \left(34 a^{5} + 11 a^{4} + 28 a^{3} + 22 a^{2} + 11 a + 36\right)\cdot 37^{4} + \left(11 a^{5} + 34 a^{4} + 3 a^{3} + 6 a^{2} + 20 a + 26\right)\cdot 37^{5} + \left(8 a^{5} + 5 a^{4} + 36 a^{3} + 28 a^{2} + 22 a + 14\right)\cdot 37^{6} + \left(29 a^{5} + 33 a^{4} + 6 a^{3} + 10 a^{2} + 9 a + 1\right)\cdot 37^{7} + \left(11 a^{5} + 24 a^{4} + 16 a^{3} + 6 a^{2} + 9 a + 19\right)\cdot 37^{8} + \left(19 a^{5} + 2 a^{4} + 32 a^{3} + 3 a^{2} + 15 a + 34\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 5 }$ $=$ $$7 a^{5} + 29 a^{4} + 36 a^{3} + 20 a^{2} + 12 a + 26 + \left(29 a^{5} + 22 a^{4} + 3 a^{3} + 13 a^{2} + 34 a + 16\right)\cdot 37 + \left(8 a^{5} + 9 a^{4} + 33 a^{3} + 15 a^{2} + 30 a + 24\right)\cdot 37^{2} + \left(13 a^{5} + 6 a^{4} + 28 a^{3} + 27 a^{2} + 33 a + 29\right)\cdot 37^{3} + \left(5 a^{5} + 9 a^{4} + 33 a^{3} + 24 a^{2} + 31 a + 27\right)\cdot 37^{4} + \left(12 a^{5} + 23 a^{4} + 6 a^{3} + 27 a^{2} + 18 a + 32\right)\cdot 37^{5} + \left(28 a^{5} + 19 a^{4} + 8 a^{3} + 15 a^{2} + 18 a + 13\right)\cdot 37^{6} + \left(30 a^{5} + 3 a^{4} + 29 a^{3} + 20 a^{2} + 26 a + 12\right)\cdot 37^{7} + \left(3 a^{5} + 30 a^{4} + 5 a^{3} + 27 a^{2} + 34 a + 15\right)\cdot 37^{8} + \left(29 a^{5} + 14 a^{4} + 36 a^{3} + 26 a^{2} + 19 a + 2\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 6 }$ $=$ $$11 a^{5} + 9 a^{4} + 34 a^{3} + 35 a^{2} + 30 a + \left(24 a^{5} + 22 a^{4} + 19 a^{3} + 27 a^{2} + 34 a + 13\right)\cdot 37 + \left(19 a^{5} + 4 a^{4} + 29 a^{3} + a^{2} + 32 a + 9\right)\cdot 37^{2} + \left(27 a^{5} + 32 a^{4} + 33 a^{3} + 8 a^{2} + 36 a + 14\right)\cdot 37^{3} + \left(26 a^{5} + 25 a^{4} + 10 a^{3} + 10 a^{2} + 18 a + 1\right)\cdot 37^{4} + \left(34 a^{5} + 33 a^{4} + 32 a^{3} + 8 a^{2} + 4 a + 9\right)\cdot 37^{5} + \left(30 a^{5} + 24 a^{4} + a^{3} + 31 a^{2} + 19 a + 13\right)\cdot 37^{6} + \left(36 a^{5} + 18 a^{4} + 16 a^{3} + 16 a^{2} + 17 a + 34\right)\cdot 37^{7} + \left(26 a^{5} + 17 a^{4} + 7 a^{3} + 3 a^{2} + 36 a + 28\right)\cdot 37^{8} + \left(16 a^{5} + 26 a^{4} + 22 a^{3} + 28 a^{2} + 5 a + 26\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 7 }$ $=$ $$4 a^{5} + 6 a^{4} + 10 a^{3} + 22 a^{2} + 28 a + 24 + \left(16 a^{5} + 12 a^{4} + 27 a^{3} + 4 a^{2} + 36 a + 27\right)\cdot 37 + \left(4 a^{5} + 6 a^{4} + 8 a^{3} + 28 a^{2} + 34 a + 13\right)\cdot 37^{2} + \left(12 a^{5} + 21 a^{4} + 33 a^{3} + 14 a^{2} + 33 a + 17\right)\cdot 37^{3} + \left(18 a^{5} + 11 a^{4} + 29 a^{3} + 23 a + 23\right)\cdot 37^{4} + \left(12 a^{5} + 33 a^{4} + 32 a^{3} + 3 a^{2} + 3 a + 1\right)\cdot 37^{5} + \left(17 a^{5} + 16 a^{4} + 17 a^{3} + 9 a^{2} + 13 a + 15\right)\cdot 37^{6} + \left(22 a^{5} + 14 a^{4} + 14 a^{3} + 11 a^{2} + 30 a + 31\right)\cdot 37^{7} + \left(18 a^{5} + 30 a^{4} + 27 a^{3} + 25 a^{2} + 34 a + 35\right)\cdot 37^{8} + \left(14 a^{5} + 22 a^{4} + 35 a^{3} + 20 a^{2} + 31 a + 1\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 8 }$ $=$ $$33 a^{5} + 13 a^{4} + 26 a^{3} + 14 a^{2} + 16 a + 26 + \left(6 a^{5} + 20 a^{4} + 20 a^{3} + 18 a^{2} + 26 a + 36\right)\cdot 37 + \left(11 a^{5} + 6 a^{4} + 24 a^{3} + 26 a^{2} + 23 a + 23\right)\cdot 37^{2} + \left(5 a^{5} + 11 a^{4} + 8 a^{3} + 18 a^{2} + 36 a + 29\right)\cdot 37^{3} + \left(10 a^{5} + 4 a^{4} + 6 a^{3} + 16 a^{2} + 24 a + 32\right)\cdot 37^{4} + \left(16 a^{5} + 9 a^{4} + 11 a^{3} + 18 a^{2} + 3 a + 12\right)\cdot 37^{5} + \left(17 a^{5} + 29 a^{4} + 19 a^{3} + 5 a^{2} + 12 a + 29\right)\cdot 37^{6} + \left(30 a^{5} + 21 a^{4} + 21 a^{3} + 31 a^{2} + 21 a + 34\right)\cdot 37^{7} + \left(17 a^{5} + 22 a^{4} + 36 a^{3} + 22 a^{2} + 19 a + 14\right)\cdot 37^{8} + \left(13 a^{5} + 17 a^{4} + 5 a^{3} + a^{2} + 20 a + 14\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 9 }$ $=$ $$25 a^{5} + 24 a^{4} + 14 a^{3} + 26 a^{2} + 19 a + 35 + \left(25 a^{5} + 3 a^{4} + 20 a^{3} + 9 a^{2} + a + 8\right)\cdot 37 + \left(11 a^{5} + 26 a^{4} + 21 a^{3} + 29 a^{2} + 26 a + 3\right)\cdot 37^{2} + \left(7 a^{5} + 9 a^{4} + 26 a^{3} + 35 a^{2} + 8 a + 36\right)\cdot 37^{3} + \left(27 a^{5} + 17 a^{4} + 22 a^{3} + 29 a^{2} + 4 a + 32\right)\cdot 37^{4} + \left(18 a^{5} + 9 a^{4} + 28 a^{3} + 22 a^{2} + 16 a + 9\right)\cdot 37^{5} + \left(22 a^{5} + a^{4} + 12 a^{3} + 3 a^{2} + 22 a + 15\right)\cdot 37^{6} + \left(8 a^{5} + 8 a^{4} + 36 a^{3} + 19 a^{2} + 34 a + 35\right)\cdot 37^{7} + \left(12 a^{5} + 21 a^{4} + a^{3} + 5 a^{2} + 18 a + 4\right)\cdot 37^{8} + \left(7 a^{5} + 17 a^{4} + 34 a^{3} + 10 a^{2} + 17 a + 2\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 10 }$ $=$ $$33 a^{5} + 8 a^{4} + 6 a^{3} + 10 a^{2} + 14 a + 3 + \left(5 a^{5} + 15 a^{4} + 24 a^{3} + 26 a^{2} + 7 a + 6\right)\cdot 37 + \left(3 a^{5} + 13 a^{4} + 28 a^{3} + 33 a^{2} + a + 22\right)\cdot 37^{2} + \left(11 a^{5} + 11 a^{4} + 26 a^{3} + 7 a^{2} + 9 a + 8\right)\cdot 37^{3} + \left(16 a^{5} + 19 a^{4} + 25 a^{3} + 5 a^{2} + 12 a + 28\right)\cdot 37^{4} + \left(16 a^{5} + 2 a^{4} + 6 a^{3} + 26 a^{2} + 21 a + 6\right)\cdot 37^{5} + \left(15 a^{5} + 6 a^{4} + 11 a^{3} + 26 a^{2} + 21 a + 27\right)\cdot 37^{6} + \left(21 a^{5} + 23 a^{4} + 13 a^{3} + 24 a^{2} + 2 a + 29\right)\cdot 37^{7} + \left(4 a^{5} + 7 a^{4} + 34 a^{3} + 14 a^{2} + 35 a + 4\right)\cdot 37^{8} + \left(10 a^{5} + 18 a^{4} + 35 a^{3} + 30 a^{2} + 2 a + 20\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 11 }$ $=$ $$10 a^{5} + 6 a^{4} + 7 a^{3} + 9 a^{2} + 7 a + 26 + \left(25 a^{5} + 22 a^{4} + 8 a^{3} + 10 a^{2} + 21 a + 13\right)\cdot 37 + \left(21 a^{5} + 6 a^{4} + 25 a^{3} + 5 a^{2} + 2 a + 12\right)\cdot 37^{2} + \left(35 a^{5} + 10 a^{4} + 33 a^{3} + 24 a^{2} + 19 a + 17\right)\cdot 37^{3} + \left(34 a^{5} + 10 a^{4} + 33 a^{3} + 26 a^{2} + 9 a + 3\right)\cdot 37^{4} + \left(34 a^{5} + 17 a^{4} + 28 a^{3} + 28 a^{2} + 10 a + 28\right)\cdot 37^{5} + \left(28 a^{5} + 32 a^{4} + 33 a^{3} + 30 a^{2} + 34 a + 22\right)\cdot 37^{6} + \left(13 a^{5} + a^{4} + 3 a^{3} + 10 a^{2} + 20 a + 18\right)\cdot 37^{7} + \left(33 a^{5} + 12 a^{4} + 32 a^{3} + 18 a^{2} + 26 a + 12\right)\cdot 37^{8} + \left(9 a^{5} + 36 a^{4} + 32 a^{3} + 3 a^{2} + 11 a + 9\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 12 }$ $=$ $$14 a^{5} + 11 a^{4} + 14 a^{3} + 8 a^{2} + 9 a + 14 + \left(22 a^{5} + 17 a^{4} + 8 a^{3} + 15 a^{2} + 7 a + 27\right)\cdot 37 + \left(5 a^{5} + 8 a^{4} + 26 a^{3} + 3 a^{2} + 25 a + 29\right)\cdot 37^{2} + \left(8 a^{5} + 14 a^{3} + 35 a^{2} + 24 a + 4\right)\cdot 37^{3} + \left(26 a^{5} + 29 a^{4} + 33 a^{3} + 26 a^{2} + 15 a + 20\right)\cdot 37^{4} + \left(31 a^{5} + 6 a^{4} + 13 a^{3} + 26 a^{2} + 36 a + 2\right)\cdot 37^{5} + \left(19 a^{5} + 23 a^{4} + 26 a^{3} + 20 a^{2} + 17 a\right)\cdot 37^{6} + \left(12 a^{5} + 33 a^{4} + 5 a^{3} + 25 a^{2} + 7 a + 32\right)\cdot 37^{7} + \left(12 a^{5} + 6 a^{4} + 6 a^{3} + 36 a^{2} + 23 a + 18\right)\cdot 37^{8} + \left(24 a^{5} + 15 a^{4} + 22 a^{3} + 26 a^{2} + 23 a + 13\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 13 }$ $=$ $$30 a^{5} + 8 a^{4} + a^{3} + 17 a^{2} + 25 a + 25 + \left(7 a^{5} + 14 a^{4} + 33 a^{3} + 23 a^{2} + 2 a + 35\right)\cdot 37 + \left(28 a^{5} + 27 a^{4} + 3 a^{3} + 21 a^{2} + 6 a + 2\right)\cdot 37^{2} + \left(23 a^{5} + 30 a^{4} + 8 a^{3} + 9 a^{2} + 3 a + 25\right)\cdot 37^{3} + \left(31 a^{5} + 27 a^{4} + 3 a^{3} + 12 a^{2} + 5 a + 8\right)\cdot 37^{4} + \left(24 a^{5} + 13 a^{4} + 30 a^{3} + 9 a^{2} + 18 a + 9\right)\cdot 37^{5} + \left(8 a^{5} + 17 a^{4} + 28 a^{3} + 21 a^{2} + 18 a + 19\right)\cdot 37^{6} + \left(6 a^{5} + 33 a^{4} + 7 a^{3} + 16 a^{2} + 10 a + 22\right)\cdot 37^{7} + \left(33 a^{5} + 6 a^{4} + 31 a^{3} + 9 a^{2} + 2 a + 15\right)\cdot 37^{8} + \left(7 a^{5} + 22 a^{4} + 10 a^{2} + 17 a + 2\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 14 }$ $=$ $$31 a^{5} + 27 a^{4} + 7 a^{3} + 17 a^{2} + 9 a + 20 + \left(3 a^{5} + 28 a^{4} + 24 a^{3} + 34 a^{2} + 3 a\right)\cdot 37 + \left(18 a^{5} + 11 a^{4} + 3 a^{3} + 16 a^{2} + 10 a + 30\right)\cdot 37^{2} + \left(12 a^{5} + 32 a^{4} + 7 a^{3} + 16 a^{2} + 10 a + 34\right)\cdot 37^{3} + \left(a^{5} + 35 a^{4} + 9 a^{3} + 11 a^{2} + 33 a + 34\right)\cdot 37^{4} + \left(10 a^{5} + 11 a^{4} + 12 a^{3} + 8 a^{2} + 32 a + 12\right)\cdot 37^{5} + \left(25 a^{5} + 35 a^{4} + 19 a^{3} + 8 a^{2} + 27 a + 1\right)\cdot 37^{6} + \left(11 a^{5} + 14 a^{4} + 33 a^{3} + 15 a^{2} + 30 a + 22\right)\cdot 37^{7} + \left(22 a^{5} + 11 a^{4} + 21 a^{3} + 5 a^{2} + 19 a + 25\right)\cdot 37^{8} + \left(21 a^{5} + 8 a^{4} + 33 a^{3} + 29 a^{2} + 2 a + 9\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 15 }$ $=$ $$33 a^{5} + 31 a^{4} + 27 a^{3} + 15 a^{2} + 9 a + 34 + \left(20 a^{5} + 24 a^{4} + 9 a^{3} + 32 a^{2} + 15\right)\cdot 37 + \left(32 a^{5} + 30 a^{4} + 28 a^{3} + 8 a^{2} + 2 a + 7\right)\cdot 37^{2} + \left(24 a^{5} + 15 a^{4} + 3 a^{3} + 22 a^{2} + 3 a + 22\right)\cdot 37^{3} + \left(18 a^{5} + 25 a^{4} + 7 a^{3} + 36 a^{2} + 13 a + 21\right)\cdot 37^{4} + \left(24 a^{5} + 3 a^{4} + 4 a^{3} + 33 a^{2} + 33 a + 36\right)\cdot 37^{5} + \left(19 a^{5} + 20 a^{4} + 19 a^{3} + 27 a^{2} + 23 a + 22\right)\cdot 37^{6} + \left(14 a^{5} + 22 a^{4} + 22 a^{3} + 25 a^{2} + 6 a + 13\right)\cdot 37^{7} + \left(18 a^{5} + 6 a^{4} + 9 a^{3} + 11 a^{2} + 2 a + 7\right)\cdot 37^{8} + \left(22 a^{5} + 14 a^{4} + a^{3} + 16 a^{2} + 5 a + 13\right)\cdot 37^{9} +O(37^{10})$$ $r_{ 16 }$ $=$ $$25 a^{5} + 18 a^{4} + 3 a^{3} + 10 a^{2} + 15 a + 35 + \left(31 a^{5} + 20 a^{4} + 5 a^{3} + 22 a^{2}\right)\cdot 37 + \left(19 a^{5} + 12 a^{4} + 11 a^{3} + 23 a^{2} + 17 a + 28\right)\cdot 37^{2} + \left(20 a^{4} + 36 a^{3} + 11 a^{2} + 24 a + 17\right)\cdot 37^{3} + \left(12 a^{5} + 4 a^{4} + 22 a^{3} + 23 a^{2} + 6 a + 9\right)\cdot 37^{4} + \left(6 a^{5} + 15 a^{4} + 5 a^{2} + 13 a + 27\right)\cdot 37^{5} + \left(9 a^{5} + 29 a^{4} + 8 a^{3} + 17 a^{2} + 11 a + 1\right)\cdot 37^{6} + \left(27 a^{5} + 25 a^{4} + 27 a^{3} + 11 a^{2} + 24 a + 9\right)\cdot 37^{7} + \left(19 a^{5} + 27 a^{4} + 22 a^{3} + 36 a^{2} + 2 a + 25\right)\cdot 37^{8} + \left(15 a^{5} + 3 a^{4} + 31 a^{3} + 21 a^{2} + 6 a + 14\right)\cdot 37^{9} +O(37^{10})$$

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

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

## Character values on conjugacy classes

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

The blue line marks the conjugacy class containing complex conjugation.