Basic invariants
Dimension: | $4$ |
Group: | $C_2^3:S_4$ |
Conductor: | \(5744\)\(\medspace = 2^{4} \cdot 359 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.0.8248384.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $C_2^3:S_4$ |
Parity: | even |
Determinant: | 1.1436.2t1.a.a |
Projective image: | $C_2^2:S_4$ |
Projective stem field: | Galois closure of 8.0.68035838611456.3 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - x^{7} - x^{6} + 3x^{5} + 2x^{4} - 3x^{3} - x^{2} + x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$: \( x^{3} + x + 28 \)
Roots:
$r_{ 1 }$ | $=$ | \( 26 + 27\cdot 31^{2} + 6\cdot 31^{3} + 4\cdot 31^{4} + 28\cdot 31^{5} + 13\cdot 31^{6} + 28\cdot 31^{7} + 23\cdot 31^{8} + 18\cdot 31^{9} +O(31^{10})\) |
$r_{ 2 }$ | $=$ | \( 13 a^{2} + 12 a + 26 + \left(27 a^{2} + 16 a + 19\right)\cdot 31 + \left(10 a^{2} + 4 a + 10\right)\cdot 31^{2} + \left(15 a^{2} + 7 a + 6\right)\cdot 31^{3} + \left(25 a^{2} + 23 a + 7\right)\cdot 31^{4} + \left(16 a^{2} + 4 a + 7\right)\cdot 31^{5} + \left(4 a^{2} + 5 a + 8\right)\cdot 31^{6} + \left(3 a^{2} + 15 a + 11\right)\cdot 31^{7} + \left(3 a^{2} + 20 a + 12\right)\cdot 31^{8} + \left(12 a^{2} + 27 a + 1\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 3 }$ | $=$ | \( 9 a^{2} + 5 a + 13 + \left(16 a^{2} + 4 a + 12\right)\cdot 31 + \left(19 a^{2} + 16\right)\cdot 31^{2} + \left(2 a^{2} + 6 a + 18\right)\cdot 31^{3} + \left(3 a^{2} + 12 a + 2\right)\cdot 31^{4} + \left(12 a^{2} + 28 a + 4\right)\cdot 31^{5} + \left(3 a + 26\right)\cdot 31^{6} + \left(3 a^{2} + 24 a\right)\cdot 31^{7} + \left(18 a^{2} + 22 a + 12\right)\cdot 31^{8} + \left(25 a^{2} + 9 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 4 }$ | $=$ | \( 15 a^{2} + 25 a + 7 + \left(26 a^{2} + 30 a + 16\right)\cdot 31 + \left(24 a^{2} + 8 a + 29\right)\cdot 31^{2} + \left(2 a^{2} + 21 a + 8\right)\cdot 31^{3} + \left(30 a^{2} + 4 a + 7\right)\cdot 31^{4} + \left(28 a^{2} + 15 a + 19\right)\cdot 31^{5} + \left(17 a^{2} + 4 a + 28\right)\cdot 31^{6} + \left(19 a^{2} + 29 a + 16\right)\cdot 31^{7} + \left(19 a^{2} + 27 a + 10\right)\cdot 31^{8} + \left(16 a^{2} + 25 a + 10\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 5 }$ | $=$ | \( 20 a^{2} + 30 a + \left(27 a^{2} + 17 a + 17\right)\cdot 31 + \left(21 a^{2} + 29 a + 27\right)\cdot 31^{2} + \left(10 a^{2} + 21 a + 3\right)\cdot 31^{3} + \left(14 a^{2} + 3 a + 7\right)\cdot 31^{4} + \left(6 a^{2} + 11 a + 4\right)\cdot 31^{5} + \left(27 a^{2} + 6 a + 14\right)\cdot 31^{6} + \left(12 a^{2} + 17 a + 12\right)\cdot 31^{7} + \left(30 a^{2} + 2 a + 7\right)\cdot 31^{8} + \left(19 a^{2} + 10 a + 2\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 6 }$ | $=$ | \( 25 + 29\cdot 31 + 16\cdot 31^{2} + 14\cdot 31^{3} + 31^{4} + 15\cdot 31^{5} + 13\cdot 31^{6} + 25\cdot 31^{7} + 14\cdot 31^{8} + 3\cdot 31^{9} +O(31^{10})\) |
$r_{ 7 }$ | $=$ | \( 9 a^{2} + 14 a + 13 + \left(18 a^{2} + 10 a + 3\right)\cdot 31 + \left(26 a + 14\right)\cdot 31^{2} + \left(13 a^{2} + 17 a + 25\right)\cdot 31^{3} + \left(2 a^{2} + 26 a + 22\right)\cdot 31^{4} + \left(2 a^{2} + 28 a + 7\right)\cdot 31^{5} + \left(26 a^{2} + 21 a + 12\right)\cdot 31^{6} + \left(24 a^{2} + 22 a + 15\right)\cdot 31^{7} + \left(9 a^{2} + 18 a + 6\right)\cdot 31^{8} + \left(24 a^{2} + 24 a + 30\right)\cdot 31^{9} +O(31^{10})\) |
$r_{ 8 }$ | $=$ | \( 27 a^{2} + 7 a + 15 + \left(7 a^{2} + 13 a + 24\right)\cdot 31 + \left(15 a^{2} + 23 a + 12\right)\cdot 31^{2} + \left(17 a^{2} + 18 a + 8\right)\cdot 31^{3} + \left(17 a^{2} + 22 a + 9\right)\cdot 31^{4} + \left(26 a^{2} + 4 a + 7\right)\cdot 31^{5} + \left(16 a^{2} + 20 a + 7\right)\cdot 31^{6} + \left(29 a^{2} + 15 a + 13\right)\cdot 31^{7} + \left(11 a^{2} + 5\right)\cdot 31^{8} + \left(25 a^{2} + 26 a + 16\right)\cdot 31^{9} +O(31^{10})\) |
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
Cycle notation |
Character values on conjugacy classes
Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,6)(2,5)(3,4)(7,8)$ | $-4$ |
$6$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
$6$ | $2$ | $(1,8)(2,4)(3,5)(6,7)$ | $0$ |
$6$ | $2$ | $(1,6)(7,8)$ | $0$ |
$12$ | $2$ | $(1,3)(4,6)$ | $2$ |
$12$ | $2$ | $(1,8)(2,5)(3,4)(6,7)$ | $-2$ |
$32$ | $3$ | $(1,7,3)(4,6,8)$ | $1$ |
$12$ | $4$ | $(1,7,6,8)(2,3,5,4)$ | $0$ |
$24$ | $4$ | $(1,2,7,4)(3,6,5,8)$ | $0$ |
$24$ | $4$ | $(1,2,8,4)(3,6,5,7)$ | $0$ |
$24$ | $4$ | $(1,7,6,8)(2,5)$ | $0$ |
$32$ | $6$ | $(1,4,7,6,3,8)(2,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.