Basic invariants
Dimension: | $2$ |
Group: | $C_7 \wr C_2$ |
Conductor: | \(19343\)\(\medspace = 23 \cdot 29^{2} \) |
Artin stem field: | Galois closure of 14.0.2025269579809849487.1 |
Galois orbit size: | $6$ |
Smallest permutation container: | $C_7 \wr C_2$ |
Parity: | odd |
Determinant: | 1.667.14t1.a.e |
Projective image: | $D_7$ |
Projective stem field: | Galois closure of 7.1.7237215346607.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{14} - 3 x^{13} + 11 x^{12} - 26 x^{11} + 65 x^{10} - 64 x^{9} + 69 x^{8} - 51 x^{7} + 82 x^{6} + \cdots + 8 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{7} + x + 28 \)
Roots:
$r_{ 1 }$ | $=$ | \( 2 a^{6} + 22 a^{5} + 29 a^{4} + 7 a^{3} + 9 a^{2} + 4 a + 3 + \left(24 a^{6} + 22 a^{5} + 29 a^{4} + 7 a^{3} + 9 a^{2} + 10 a + 30\right)\cdot 31 + \left(22 a^{6} + 25 a^{5} + 21 a^{4} + 4 a^{3} + 17 a^{2} + 3 a + 6\right)\cdot 31^{2} + \left(13 a^{6} + 11 a^{5} + 24 a^{4} + 13 a^{3} + 7 a^{2} + 23 a + 15\right)\cdot 31^{3} + \left(18 a^{6} + 15 a^{5} + 7 a^{4} + 27 a^{3} + 4 a^{2} + 20 a + 28\right)\cdot 31^{4} + \left(11 a^{6} + 13 a^{4} + 16 a^{3} + 16 a^{2} + 19 a + 10\right)\cdot 31^{5} + \left(30 a^{6} + 3 a^{5} + 23 a^{4} + 13 a^{3} + 8 a^{2} + 2 a\right)\cdot 31^{6} + \left(25 a^{6} + 16 a^{5} + 22 a^{4} + 26 a^{3} + 12 a^{2} + 12 a + 7\right)\cdot 31^{7} + \left(17 a^{6} + 25 a^{5} + 9 a^{4} + 26 a^{3} + 4 a^{2} + 23 a + 3\right)\cdot 31^{8} + \left(23 a^{6} + 21 a^{5} + 3 a^{4} + 21 a^{3} + 17 a^{2} + 17 a + 5\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 3 a^{6} + 21 a^{5} + a^{4} + 25 a^{3} + 23 a^{2} + 24 a + 26 + \left(13 a^{6} + 2 a^{5} + 6 a^{4} + 7 a^{3} + 14 a^{2} + 21 a + 11\right)\cdot 31 + \left(26 a^{6} + 2 a^{5} + 12 a^{4} + 28 a^{3} + 19 a^{2} + 17 a + 23\right)\cdot 31^{2} + \left(8 a^{6} + 2 a^{5} + 29 a^{4} + 12 a^{3} + 23 a^{2} + 24 a + 28\right)\cdot 31^{3} + \left(5 a^{6} + 30 a^{5} + 15 a^{4} + 9 a^{3} + a^{2} + 3\right)\cdot 31^{4} + \left(7 a^{6} + 10 a^{5} + 21 a^{4} + 11 a^{3} + 26 a^{2} + 20 a + 7\right)\cdot 31^{5} + \left(26 a^{6} + 25 a^{5} + 7 a^{4} + 30 a^{3} + 21 a^{2} + 4 a + 1\right)\cdot 31^{6} + \left(18 a^{6} + 7 a^{5} + 20 a^{4} + 20 a^{3} + 15 a + 23\right)\cdot 31^{7} + \left(22 a^{6} + 5 a^{5} + 19 a^{4} + 23 a^{3} + 2 a^{2} + 24 a + 11\right)\cdot 31^{8} + \left(29 a^{6} + 19 a^{5} + 23 a^{4} + 20 a^{3} + 27 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 4 a^{6} + 6 a^{5} + 29 a^{4} + 4 a^{3} + 29 a^{2} + 10 a + 7 + \left(30 a^{6} + 8 a^{5} + 26 a^{4} + 16 a^{3} + 30 a^{2} + 25 a + 25\right)\cdot 31 + \left(a^{6} + 8 a^{5} + 25 a^{4} + 25 a^{3} + 3 a^{2} + 9 a + 9\right)\cdot 31^{2} + \left(30 a^{6} + 23 a^{5} + 18 a^{4} + 30 a^{3} + 9 a^{2} + 29 a + 22\right)\cdot 31^{3} + \left(10 a^{6} + 10 a^{5} + 8 a^{4} + 23 a^{3} + 21 a^{2} + a + 27\right)\cdot 31^{4} + \left(a^{6} + 25 a^{5} + 30 a^{4} + 15 a^{3} + 8 a^{2} + 16 a + 17\right)\cdot 31^{5} + \left(9 a^{6} + 14 a^{5} + 2 a^{4} + 19 a^{3} + 6 a^{2} + 6 a + 24\right)\cdot 31^{6} + \left(26 a^{6} + 21 a^{5} + 19 a^{4} + 12 a^{3} + 13 a^{2} + 4 a + 6\right)\cdot 31^{7} + \left(10 a^{6} + 8 a^{5} + 3 a^{4} + 3 a^{3} + 25 a^{2} + 5 a + 30\right)\cdot 31^{8} + \left(15 a^{6} + 23 a^{5} + 15 a^{4} + 18 a^{3} + a^{2} + 25 a + 14\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 5 a^{6} + 20 a^{5} + 22 a^{4} + 5 a^{3} + 17 a^{2} + 28 a + 10 + \left(11 a^{6} + 23 a^{5} + 29 a^{4} + 25 a^{3} + 29 a + 1\right)\cdot 31 + \left(10 a^{6} + 14 a^{5} + 19 a^{4} + 7 a^{3} + 28 a^{2} + 4 a + 14\right)\cdot 31^{2} + \left(6 a^{6} + 17 a^{5} + 9 a^{4} + 9 a^{3} + 2 a^{2} + 29 a + 4\right)\cdot 31^{3} + \left(20 a^{6} + 6 a^{5} + 26 a^{4} + 13 a^{3} + 30 a^{2} + 20 a + 21\right)\cdot 31^{4} + \left(10 a^{6} + 13 a^{5} + 16 a^{4} + 6 a^{3} + 4 a^{2} + 19 a + 5\right)\cdot 31^{5} + \left(14 a^{6} + 4 a^{5} + 29 a^{4} + 18 a^{3} + 24 a^{2} + 16 a + 13\right)\cdot 31^{6} + \left(a^{6} + 21 a^{5} + 25 a^{4} + 10 a^{3} + 7 a^{2} + 18 a + 21\right)\cdot 31^{7} + \left(3 a^{6} + 3 a^{5} + 6 a^{4} + 17 a^{3} + 12 a^{2} + 22 a + 12\right)\cdot 31^{8} + \left(2 a^{6} + 2 a^{5} + 30 a^{4} + 4 a^{3} + 20 a^{2} + 24 a + 4\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 6 a^{6} + 28 a^{5} + 13 a^{4} + 5 a^{3} + 14 a^{2} + 19 a + 22 + \left(19 a^{6} + a^{5} + 15 a^{4} + 25 a^{3} + 4 a^{2} + 28 a + 24\right)\cdot 31 + \left(25 a^{6} + 26 a^{5} + 27 a^{4} + 14 a^{3} + 7 a^{2} + 11 a + 16\right)\cdot 31^{2} + \left(16 a^{6} + 23 a^{5} + 10 a^{4} + 26 a^{3} + 28 a^{2} + 19 a + 6\right)\cdot 31^{3} + \left(15 a^{6} + 27 a^{5} + 24 a^{4} + 25 a^{3} + 11 a^{2} + 18 a + 27\right)\cdot 31^{4} + \left(17 a^{6} + 7 a^{5} + 6 a^{4} + 23 a^{3} + 25 a^{2} + 21 a + 22\right)\cdot 31^{5} + \left(18 a^{6} + 3 a^{5} + 8 a^{4} + 11 a^{3} + 9 a^{2} + 28 a + 1\right)\cdot 31^{6} + \left(29 a^{6} + 16 a^{5} + 3 a^{4} + 21 a^{3} + 16 a^{2} + 24 a + 14\right)\cdot 31^{7} + \left(8 a^{6} + 11 a^{5} + 14 a^{4} + 23 a^{3} + 7 a^{2} + 23 a + 6\right)\cdot 31^{8} + \left(5 a^{6} + 9 a^{5} + 30 a^{4} + 22 a^{3} + 25 a^{2} + 24 a + 15\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 12 a^{6} + 28 a^{5} + 11 a^{4} + 21 a^{3} + 13 a^{2} + 2 a + 5 + \left(14 a^{6} + 16 a^{5} + 3 a^{4} + 28 a^{3} + 17 a^{2} + 16 a + 16\right)\cdot 31 + \left(30 a^{6} + 26 a^{5} + 12 a^{4} + 15 a^{3} + 23 a^{2} + 16 a + 16\right)\cdot 31^{2} + \left(a^{6} + a^{5} + 11 a^{4} + 5 a^{3} + 10 a^{2} + 10 a + 11\right)\cdot 31^{3} + \left(a^{5} + 21 a^{4} + 11 a^{3} + 21 a^{2} + 10 a + 18\right)\cdot 31^{4} + \left(2 a^{6} + 17 a^{5} + 7 a^{4} + 17 a^{3} + 17 a^{2} + 17 a + 9\right)\cdot 31^{5} + \left(9 a^{6} + 24 a^{5} + 6 a^{4} + 28 a^{3} + 24 a^{2} + 27 a + 11\right)\cdot 31^{6} + \left(28 a^{6} + 13 a^{5} + 20 a^{4} + 19 a^{3} + 20 a^{2} + 18 a + 17\right)\cdot 31^{7} + \left(24 a^{6} + 11 a^{5} + 30 a^{4} + 28 a^{3} + 4 a^{2} + 7 a + 24\right)\cdot 31^{8} + \left(17 a^{6} + 16 a^{5} + 5 a^{4} + 18 a^{3} + 12 a^{2} + 22 a + 3\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 15 a^{6} + 11 a^{5} + 24 a^{4} + 14 a^{2} + 23 + \left(24 a^{6} + 10 a^{5} + 2 a^{4} + 15 a^{3} + 25 a^{2} + 10 a + 12\right)\cdot 31 + \left(7 a^{6} + 10 a^{5} + 7 a^{4} + a^{3} + 18 a^{2} + 21 a + 7\right)\cdot 31^{2} + \left(11 a^{6} + 16 a^{5} + 8 a^{4} + 12 a^{3} + a^{2} + 10 a + 4\right)\cdot 31^{3} + \left(5 a^{6} + 7 a^{5} + 22 a^{3} + 17 a^{2} + 5 a + 26\right)\cdot 31^{4} + \left(25 a^{6} + 8 a^{5} + 26 a^{4} + 15 a^{3} + 6 a^{2} + 21 a + 4\right)\cdot 31^{5} + \left(30 a^{6} + 12 a^{5} + 24 a^{4} + 19 a^{3} + 25 a^{2} + 10 a + 5\right)\cdot 31^{6} + \left(11 a^{6} + 4 a^{5} + 9 a^{4} + 24 a^{3} + 19 a^{2} + 21 a + 17\right)\cdot 31^{7} + \left(13 a^{5} + 3 a^{4} + 24 a^{3} + 10 a^{2} + 4 a + 1\right)\cdot 31^{8} + \left(11 a^{6} + 29 a^{5} + 7 a^{4} + 26 a^{3} + 16 a + 12\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 18 a^{6} + 11 a^{5} + 3 a^{4} + 22 a^{3} + 6 a^{2} + 15 a + 30 + \left(2 a^{6} + 6 a^{5} + 10 a^{4} + 17 a^{3} + 7 a^{2} + 24\right)\cdot 31 + \left(4 a^{6} + 10 a^{5} + 27 a^{4} + 19 a^{2} + 27 a + 30\right)\cdot 31^{2} + \left(10 a^{6} + 28 a^{5} + 2 a^{4} + 9 a^{3} + 11 a^{2} + 8 a + 20\right)\cdot 31^{3} + \left(7 a^{6} + a^{5} + 2 a^{4} + 18 a^{3} + 8 a^{2} + 8 a + 5\right)\cdot 31^{4} + \left(21 a^{6} + 16 a^{5} + 10 a^{4} + 5 a^{3} + 7 a^{2} + 19\right)\cdot 31^{5} + \left(5 a^{6} + 18 a^{5} + 24 a^{4} + 8 a^{3} + 4 a + 14\right)\cdot 31^{6} + \left(28 a^{6} + 19 a^{5} + 22 a^{3} + 9 a^{2} + 17 a + 4\right)\cdot 31^{7} + \left(15 a^{6} + a^{5} + 24 a^{4} + 15 a^{3} + 21 a^{2} + 10 a + 28\right)\cdot 31^{8} + \left(5 a^{6} + 24 a^{5} + 4 a^{4} + 13 a^{3} + 29 a^{2} + 30 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 9 }$ | $=$ | \( 23 a^{6} + 7 a^{5} + 13 a^{4} + 2 a^{3} + 2 a^{2} + 5 a + 10 + \left(6 a^{6} + 23 a^{5} + 3 a^{4} + 21 a^{3} + 16 a^{2} + 13 a + 27\right)\cdot 31 + \left(24 a^{6} + 18 a^{5} + 15 a^{4} + 2 a^{3} + 24 a^{2} + 18 a + 28\right)\cdot 31^{2} + \left(2 a^{6} + 12 a^{5} + 27 a^{4} + 2 a^{3} + 2 a^{2} + 13 a + 29\right)\cdot 31^{3} + \left(24 a^{6} + 9 a^{5} + 28 a^{4} + 12 a^{3} + 2 a^{2} + 9 a + 7\right)\cdot 31^{4} + \left(24 a^{6} + 17 a^{5} + 29 a^{4} + 15 a^{3} + 8 a^{2} + 7 a + 20\right)\cdot 31^{5} + \left(26 a^{6} + 5 a^{5} + 19 a^{4} + 7 a^{3} + 7 a^{2} + 12 a + 17\right)\cdot 31^{6} + \left(27 a^{6} + 8 a^{5} + 23 a^{4} + 14 a^{3} + 28 a^{2} + 19 a + 12\right)\cdot 31^{7} + \left(14 a^{6} + 21 a^{5} + 21 a^{4} + 23 a^{3} + 28 a^{2} + 22 a + 20\right)\cdot 31^{8} + \left(17 a^{6} + 2 a^{5} + 9 a^{4} + 3 a^{3} + 26 a^{2} + 20 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 10 }$ | $=$ | \( 23 a^{6} + 18 a^{5} + 27 a^{4} + 12 a^{3} + 11 a^{2} + 27 a + 10 + \left(22 a^{6} + 3 a^{5} + 28 a^{4} + a^{3} + 6 a^{2} + 11 a + 23\right)\cdot 31 + \left(15 a^{6} + a^{5} + 19 a^{4} + 27 a^{3} + 24 a^{2} + 17 a + 12\right)\cdot 31^{2} + \left(19 a^{6} + 17 a^{5} + 9 a^{4} + 24 a^{3} + 21 a + 4\right)\cdot 31^{3} + \left(14 a^{6} + 26 a^{5} + 30 a^{3} + 3 a^{2} + 18 a + 13\right)\cdot 31^{4} + \left(4 a^{6} + 19 a^{5} + a^{4} + 28 a^{3} + 9 a^{2} + 2 a + 29\right)\cdot 31^{5} + \left(23 a^{6} + 12 a^{5} + 6 a^{4} + 2 a^{3} + 5 a^{2} + 25 a + 18\right)\cdot 31^{6} + \left(9 a^{6} + 29 a^{5} + 26 a^{4} + 5 a^{3} + 2 a^{2} + a + 23\right)\cdot 31^{7} + \left(28 a^{6} + 30 a^{5} + 7 a^{4} + 8 a^{3} + 8 a^{2} + 18 a + 9\right)\cdot 31^{8} + \left(6 a^{6} + 27 a^{5} + 4 a^{4} + 23 a^{3} + a^{2} + 23 a + 3\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 11 }$ | $=$ | \( 23 a^{6} + 19 a^{5} + 20 a^{4} + 6 a^{3} + 3 a + 21 + \left(30 a^{6} + 12 a^{5} + 16 a^{4} + 9 a^{3} + 30 a^{2} + 11 a + 13\right)\cdot 31 + \left(9 a^{6} + 22 a^{5} + 7 a^{4} + 5 a^{3} + 30 a^{2} + 29 a\right)\cdot 31^{2} + \left(27 a^{6} + 29 a^{5} + 8 a^{4} + 23 a^{3} + 17 a^{2} + 26 a + 18\right)\cdot 31^{3} + \left(7 a^{6} + 17 a^{5} + 8 a^{4} + a^{3} + a^{2} + 19 a + 10\right)\cdot 31^{4} + \left(a^{6} + 9 a^{5} + 9 a^{4} + 2 a^{3} + 18 a^{2} + 30 a + 15\right)\cdot 31^{5} + \left(5 a^{6} + 4 a^{5} + 24 a^{4} + 24 a^{3} + 15 a^{2} + 18\right)\cdot 31^{6} + \left(11 a^{5} + 28 a^{4} + 20 a^{3} + 8 a^{2} + 23 a + 2\right)\cdot 31^{7} + \left(15 a^{6} + 24 a^{5} + 9 a^{4} + 28 a^{3} + 27 a^{2} + 13 a + 14\right)\cdot 31^{8} + \left(2 a^{6} + 13 a^{5} + 15 a^{4} + 30 a^{3} + 5 a^{2} + 8 a + 22\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 12 }$ | $=$ | \( 27 a^{6} + 12 a^{5} + 20 a^{4} + 17 a^{3} + 5 a^{2} + a + 9 + \left(5 a^{6} + 19 a^{5} + 28 a^{4} + 15 a^{3} + 19 a^{2} + a + 13\right)\cdot 31 + \left(2 a^{6} + 23 a^{5} + 24 a^{4} + 22 a^{3} + 27 a^{2} + 3 a + 14\right)\cdot 31^{2} + \left(23 a^{6} + 21 a^{5} + 25 a^{4} + 15 a^{3} + 6 a^{2} + 26 a + 7\right)\cdot 31^{3} + \left(23 a^{6} + 3 a^{5} + 6 a^{4} + 28 a^{3} + 20 a + 3\right)\cdot 31^{4} + \left(27 a^{5} + 16 a^{4} + 16 a^{3} + 13 a^{2} + 25 a + 4\right)\cdot 31^{5} + \left(9 a^{6} + 29 a^{5} + 15 a^{4} + 25 a^{3} + 4 a + 20\right)\cdot 31^{6} + \left(13 a^{6} + 25 a^{5} + 18 a^{4} + 3 a^{3} + 2 a^{2} + 16 a + 26\right)\cdot 31^{7} + \left(16 a^{6} + 3 a^{5} + 17 a^{4} + 14 a^{3} + 21 a^{2} + 12\right)\cdot 31^{8} + \left(27 a^{6} + 18 a^{5} + 26 a^{4} + 22 a^{3} + 12 a^{2} + 2 a + 25\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 13 }$ | $=$ | \( 27 a^{6} + 20 a^{5} + 25 a^{4} + 28 a^{3} + 24 a^{2} + 19 a + 20 + \left(17 a^{6} + 14 a^{5} + 28 a^{4} + 10 a^{3} + 5 a^{2} + 9 a + 2\right)\cdot 31 + \left(11 a^{6} + 7 a^{5} + 27 a^{4} + 14 a^{3} + 21 a^{2} + 20 a + 15\right)\cdot 31^{2} + \left(15 a^{6} + 18 a^{5} + 9 a^{4} + 13 a^{3} + 27 a^{2} + 25\right)\cdot 31^{3} + \left(28 a^{6} + 13 a^{5} + a^{4} + 29 a^{2} + 17 a + 23\right)\cdot 31^{4} + \left(15 a^{6} + 3 a^{5} + 27 a^{4} + 4 a^{3} + 13 a^{2} + 12 a + 5\right)\cdot 31^{5} + \left(11 a^{6} + 25 a^{5} + 20 a^{4} + 10 a^{3} + 28 a^{2} + 22 a + 15\right)\cdot 31^{6} + \left(6 a^{6} + 12 a^{5} + 15 a^{4} + 29 a^{3} + 3 a^{2} + 16 a + 3\right)\cdot 31^{7} + \left(18 a^{6} + 19 a^{5} + 19 a^{4} + 17 a^{3} + 15 a^{2} + 24 a + 30\right)\cdot 31^{8} + \left(18 a^{6} + 13 a^{5} + 8 a^{4} + 5 a^{3} + 19 a^{2} + 29 a + 22\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 14 }$ | $=$ | \( 29 a^{6} + 25 a^{5} + 11 a^{4} + a^{3} + 19 a^{2} + 29 a + 24 + \left(24 a^{6} + 19 a^{5} + 17 a^{4} + 16 a^{3} + 29 a^{2} + 27 a + 20\right)\cdot 31 + \left(23 a^{6} + 19 a^{5} + 29 a^{4} + 15 a^{3} + 12 a^{2} + 15 a + 19\right)\cdot 31^{2} + \left(29 a^{6} + 23 a^{5} + 19 a^{4} + 18 a^{3} + 3 a^{2} + 3 a + 17\right)\cdot 31^{3} + \left(3 a^{6} + 13 a^{5} + 2 a^{4} + 22 a^{3} + 2 a^{2} + 13 a + 30\right)\cdot 31^{4} + \left(11 a^{6} + 9 a^{5} + a^{4} + 5 a^{3} + 11 a^{2} + 2 a + 12\right)\cdot 31^{5} + \left(28 a^{6} + 2 a^{5} + 3 a^{4} + 28 a^{3} + 8 a^{2} + 19 a + 23\right)\cdot 31^{6} + \left(19 a^{6} + 9 a^{5} + 13 a^{4} + 15 a^{3} + 10 a^{2} + 7 a + 5\right)\cdot 31^{7} + \left(19 a^{6} + 5 a^{5} + 28 a^{4} + 22 a^{3} + 28 a^{2} + 15 a + 11\right)\cdot 31^{8} + \left(2 a^{6} + 26 a^{5} + 14 a^{3} + 12 a^{2} + 5 a + 17\right)\cdot 31^{9} +O(31^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 14 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 14 }$ | Character value |
$1$ | $1$ | $()$ | $2$ |
$7$ | $2$ | $(1,3)(2,14)(4,12)(5,11)(6,7)(8,9)(10,13)$ | $0$ |
$1$ | $7$ | $(1,11,13,7,4,2,8)(3,5,10,6,12,14,9)$ | $2 \zeta_{7}^{5}$ |
$1$ | $7$ | $(1,13,4,8,11,7,2)(3,10,12,9,5,6,14)$ | $2 \zeta_{7}^{3}$ |
$1$ | $7$ | $(1,7,8,13,2,11,4)(3,6,9,10,14,5,12)$ | $2 \zeta_{7}$ |
$1$ | $7$ | $(1,4,11,2,13,8,7)(3,12,5,14,10,9,6)$ | $-2 \zeta_{7}^{5} - 2 \zeta_{7}^{4} - 2 \zeta_{7}^{3} - 2 \zeta_{7}^{2} - 2 \zeta_{7} - 2$ |
$1$ | $7$ | $(1,2,7,11,8,4,13)(3,14,6,5,9,12,10)$ | $2 \zeta_{7}^{4}$ |
$1$ | $7$ | $(1,8,2,4,7,13,11)(3,9,14,12,6,10,5)$ | $2 \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,8,2,4,7,13,11)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$2$ | $7$ | $(1,2,7,11,8,4,13)$ | $-\zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$2$ | $7$ | $(1,4,11,2,13,8,7)$ | $\zeta_{7}^{4} + \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,7,8,13,2,11,4)$ | $\zeta_{7}^{5} + \zeta_{7}^{3}$ |
$2$ | $7$ | $(1,13,4,8,11,7,2)$ | $\zeta_{7}^{2} + \zeta_{7}$ |
$2$ | $7$ | $(1,11,13,7,4,2,8)$ | $\zeta_{7}^{4} + \zeta_{7}$ |
$2$ | $7$ | $(1,2,7,11,8,4,13)(3,9,14,12,6,10,5)$ | $\zeta_{7}^{5} + \zeta_{7}$ |
$2$ | $7$ | $(1,7,8,13,2,11,4)(3,14,6,5,9,12,10)$ | $\zeta_{7}^{3} + \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,11,13,7,4,2,8)(3,12,5,14,10,9,6)$ | $\zeta_{7}^{3} + \zeta_{7}$ |
$2$ | $7$ | $(1,8,2,4,7,13,11)(3,6,9,10,14,5,12)$ | $-\zeta_{7}^{5} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7} - 1$ |
$2$ | $7$ | $(1,4,11,2,13,8,7)(3,10,12,9,5,6,14)$ | $\zeta_{7}^{5} + \zeta_{7}^{4}$ |
$2$ | $7$ | $(1,13,4,8,11,7,2)(3,5,10,6,12,14,9)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7} - 1$ |
$2$ | $7$ | $(1,4,11,2,13,8,7)(3,14,6,5,9,12,10)$ | $\zeta_{7}^{3} + 1$ |
$2$ | $7$ | $(1,11,13,7,4,2,8)(3,6,9,10,14,5,12)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - \zeta_{7}$ |
$2$ | $7$ | $(1,2,7,11,8,4,13)(3,5,10,6,12,14,9)$ | $\zeta_{7}^{2} + 1$ |
$2$ | $7$ | $(1,13,4,8,11,7,2)(3,9,14,12,6,10,5)$ | $\zeta_{7}^{5} + 1$ |
$2$ | $7$ | $(1,8,2,4,7,13,11)(3,12,5,14,10,9,6)$ | $\zeta_{7} + 1$ |
$2$ | $7$ | $(1,7,8,13,2,11,4)(3,10,12,9,5,6,14)$ | $\zeta_{7}^{4} + 1$ |
$2$ | $7$ | $(1,7,8,13,2,11,4)(3,12,5,14,10,9,6)$ | $\zeta_{7}^{5} + \zeta_{7}^{2}$ |
$2$ | $7$ | $(1,8,2,4,7,13,11)(3,5,10,6,12,14,9)$ | $\zeta_{7}^{4} + \zeta_{7}^{3}$ |
$2$ | $7$ | $(1,13,4,8,11,7,2)(3,14,6,5,9,12,10)$ | $-\zeta_{7}^{5} - \zeta_{7}^{4} - \zeta_{7}^{3} - \zeta_{7}^{2} - 1$ |
$7$ | $14$ | $(1,12,11,14,13,9,7,3,4,5,2,10,8,6)$ | $0$ |
$7$ | $14$ | $(1,14,7,5,8,12,13,3,2,6,11,9,4,10)$ | $0$ |
$7$ | $14$ | $(1,9,2,12,7,10,11,3,8,14,4,6,13,5)$ | $0$ |
$7$ | $14$ | $(1,5,13,6,4,14,8,3,11,10,7,12,2,9)$ | $0$ |
$7$ | $14$ | $(1,10,4,9,11,6,2,3,13,12,8,5,7,14)$ | $0$ |
$7$ | $14$ | $(1,6,8,10,2,5,4,3,7,9,13,14,11,12)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.