Basic invariants
Dimension: | $4$ |
Group: | $Q_8:S_4$ |
Conductor: | \(484416\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 29^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.2697228288.2 |
Galois orbit size: | $1$ |
Smallest permutation container: | $Q_8:S_4$ |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_2^2:S_4$ |
Projective stem field: | Galois closure of 8.0.32998902336.2 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 29x^{4} - 36x^{3} + 16x^{2} - 4 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 59 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 59 }$: \( x^{2} + 58x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 29 a + 29 + \left(a + 42\right)\cdot 59 + \left(2 a + 35\right)\cdot 59^{2} + \left(8 a + 43\right)\cdot 59^{3} + \left(30 a + 46\right)\cdot 59^{4} + \left(48 a + 46\right)\cdot 59^{5} + \left(3 a + 11\right)\cdot 59^{6} + \left(12 a + 43\right)\cdot 59^{7} + \left(25 a + 21\right)\cdot 59^{8} + \left(47 a + 14\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 2 }$ | $=$ | \( 29 a + 2 + \left(a + 44\right)\cdot 59 + \left(2 a + 22\right)\cdot 59^{2} + \left(8 a + 9\right)\cdot 59^{3} + \left(30 a + 49\right)\cdot 59^{4} + \left(48 a + 52\right)\cdot 59^{5} + \left(3 a + 32\right)\cdot 59^{6} + \left(12 a + 7\right)\cdot 59^{7} + \left(25 a + 24\right)\cdot 59^{8} + \left(47 a + 22\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 3 }$ | $=$ | \( 19 a + 50 + \left(46 a + 15\right)\cdot 59 + \left(43 a + 1\right)\cdot 59^{2} + \left(57 a + 52\right)\cdot 59^{3} + \left(45 a + 5\right)\cdot 59^{4} + \left(49 a + 57\right)\cdot 59^{5} + \left(22 a + 42\right)\cdot 59^{6} + \left(57 a + 41\right)\cdot 59^{7} + \left(4 a + 55\right)\cdot 59^{8} + \left(38 a + 12\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 4 }$ | $=$ | \( 11 + 18\cdot 59 + 40\cdot 59^{2} + 58\cdot 59^{3} + 38\cdot 59^{4} + 48\cdot 59^{5} + 27\cdot 59^{6} + 6\cdot 59^{7} + 11\cdot 59^{8} + 52\cdot 59^{9} +O(59^{10})\) |
$r_{ 5 }$ | $=$ | \( 49 + 40\cdot 59 + 18\cdot 59^{2} + 20\cdot 59^{4} + 10\cdot 59^{5} + 31\cdot 59^{6} + 52\cdot 59^{7} + 47\cdot 59^{8} + 6\cdot 59^{9} +O(59^{10})\) |
$r_{ 6 }$ | $=$ | \( 40 a + 10 + \left(12 a + 43\right)\cdot 59 + \left(15 a + 57\right)\cdot 59^{2} + \left(a + 6\right)\cdot 59^{3} + \left(13 a + 53\right)\cdot 59^{4} + \left(9 a + 1\right)\cdot 59^{5} + \left(36 a + 16\right)\cdot 59^{6} + \left(a + 17\right)\cdot 59^{7} + \left(54 a + 3\right)\cdot 59^{8} + \left(20 a + 46\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 7 }$ | $=$ | \( 30 a + 31 + \left(57 a + 16\right)\cdot 59 + \left(56 a + 23\right)\cdot 59^{2} + \left(50 a + 15\right)\cdot 59^{3} + \left(28 a + 12\right)\cdot 59^{4} + \left(10 a + 12\right)\cdot 59^{5} + \left(55 a + 47\right)\cdot 59^{6} + \left(46 a + 15\right)\cdot 59^{7} + \left(33 a + 37\right)\cdot 59^{8} + \left(11 a + 44\right)\cdot 59^{9} +O(59^{10})\) |
$r_{ 8 }$ | $=$ | \( 30 a + 58 + \left(57 a + 14\right)\cdot 59 + \left(56 a + 36\right)\cdot 59^{2} + \left(50 a + 49\right)\cdot 59^{3} + \left(28 a + 9\right)\cdot 59^{4} + \left(10 a + 6\right)\cdot 59^{5} + \left(55 a + 26\right)\cdot 59^{6} + \left(46 a + 51\right)\cdot 59^{7} + \left(33 a + 34\right)\cdot 59^{8} + \left(11 a + 36\right)\cdot 59^{9} +O(59^{10})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,7)(2,8)(3,6)(4,5)$ | $-4$ |
$6$ | $2$ | $(3,6)(4,5)$ | $0$ |
$12$ | $2$ | $(1,4)(2,3)(5,7)(6,8)$ | $0$ |
$24$ | $2$ | $(2,8)(3,4)(5,6)$ | $0$ |
$32$ | $3$ | $(1,3,2)(6,8,7)$ | $1$ |
$6$ | $4$ | $(1,3,7,6)(2,5,8,4)$ | $0$ |
$6$ | $4$ | $(1,5,7,4)(2,3,8,6)$ | $0$ |
$12$ | $4$ | $(1,7)(2,8)(3,4,6,5)$ | $-2$ |
$12$ | $4$ | $(3,4,6,5)$ | $2$ |
$32$ | $6$ | $(1,3,8,7,6,2)(4,5)$ | $-1$ |
$24$ | $8$ | $(1,6,5,2,7,3,4,8)$ | $0$ |
$24$ | $8$ | $(1,8,3,4,7,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.