Basic invariants
Dimension: | $8$ |
Group: | $Q_8:S_4$ |
Conductor: | \(606355001344\)\(\medspace = 2^{12} \cdot 23^{6} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.49836032.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | 24T332 |
Parity: | even |
Determinant: | 1.1.1t1.a.a |
Projective image: | $C_2^2:S_4$ |
Projective stem field: | Galois closure of 8.0.9474296896.1 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 2x^{7} + 2x^{6} - 4x^{5} + 5x^{4} - 4x^{3} + 2x^{2} - 2x + 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 43 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 43 }$: \( x^{2} + 42x + 3 \)
Roots:
$r_{ 1 }$ | $=$ | \( 5 a + 13 + \left(15 a + 17\right)\cdot 43 + \left(30 a + 41\right)\cdot 43^{2} + \left(27 a + 22\right)\cdot 43^{3} + \left(41 a + 21\right)\cdot 43^{4} + \left(25 a + 1\right)\cdot 43^{5} + \left(6 a + 5\right)\cdot 43^{6} + \left(21 a + 20\right)\cdot 43^{7} + \left(18 a + 18\right)\cdot 43^{8} + \left(9 a + 42\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 2 }$ | $=$ | \( 11 a + 41 + \left(16 a + 2\right)\cdot 43 + \left(9 a + 36\right)\cdot 43^{2} + \left(17 a + 28\right)\cdot 43^{3} + \left(24 a + 3\right)\cdot 43^{4} + \left(a + 34\right)\cdot 43^{5} + \left(30 a + 41\right)\cdot 43^{6} + \left(32 a + 35\right)\cdot 43^{7} + \left(6 a + 29\right)\cdot 43^{8} + \left(29 a + 10\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 3 }$ | $=$ | \( 6 a + 7 + \left(37 a + 27\right)\cdot 43 + 2\cdot 43^{2} + \left(8 a + 17\right)\cdot 43^{3} + \left(8 a + 33\right)\cdot 43^{4} + \left(13 a + 20\right)\cdot 43^{5} + \left(28 a + 35\right)\cdot 43^{6} + \left(4 a + 4\right)\cdot 43^{7} + \left(5 a + 29\right)\cdot 43^{8} + 15\cdot 43^{9} +O(43^{10})\) |
$r_{ 4 }$ | $=$ | \( 12 + 18\cdot 43 + 6\cdot 43^{2} + 9\cdot 43^{3} + 8\cdot 43^{4} + 18\cdot 43^{5} + 3\cdot 43^{7} + 13\cdot 43^{8} + 28\cdot 43^{9} +O(43^{10})\) |
$r_{ 5 }$ | $=$ | \( 18 + 12\cdot 43 + 33\cdot 43^{2} + 12\cdot 43^{3} + 25\cdot 43^{4} + 31\cdot 43^{5} + 25\cdot 43^{6} + 10\cdot 43^{7} + 32\cdot 43^{8} + 40\cdot 43^{9} +O(43^{10})\) |
$r_{ 6 }$ | $=$ | \( 38 a + 18 + \left(27 a + 27\right)\cdot 43 + \left(12 a + 13\right)\cdot 43^{2} + \left(15 a + 20\right)\cdot 43^{3} + \left(a + 35\right)\cdot 43^{4} + \left(17 a + 28\right)\cdot 43^{5} + \left(36 a + 28\right)\cdot 43^{6} + \left(21 a + 34\right)\cdot 43^{7} + \left(24 a + 15\right)\cdot 43^{8} + \left(33 a + 33\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 7 }$ | $=$ | \( 32 a + 9 + \left(26 a + 8\right)\cdot 43 + \left(33 a + 29\right)\cdot 43^{2} + \left(25 a + 36\right)\cdot 43^{3} + \left(18 a + 10\right)\cdot 43^{4} + \left(41 a + 11\right)\cdot 43^{5} + \left(12 a + 27\right)\cdot 43^{6} + \left(10 a + 38\right)\cdot 43^{7} + \left(36 a + 3\right)\cdot 43^{8} + \left(13 a + 33\right)\cdot 43^{9} +O(43^{10})\) |
$r_{ 8 }$ | $=$ | \( 37 a + 13 + \left(5 a + 15\right)\cdot 43 + \left(42 a + 9\right)\cdot 43^{2} + \left(34 a + 24\right)\cdot 43^{3} + \left(34 a + 33\right)\cdot 43^{4} + \left(29 a + 25\right)\cdot 43^{5} + \left(14 a + 7\right)\cdot 43^{6} + \left(38 a + 24\right)\cdot 43^{7} + \left(37 a + 29\right)\cdot 43^{8} + \left(42 a + 10\right)\cdot 43^{9} +O(43^{10})\) |
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 |
$1$ | $1$ | $()$ | $8$ |
$1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-8$ |
$6$ | $2$ | $(1,8)(2,7)$ | $0$ |
$12$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
$24$ | $2$ | $(1,5)(2,7)(4,8)$ | $0$ |
$32$ | $3$ | $(2,6,4)(3,5,7)$ | $-1$ |
$6$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
$6$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
$12$ | $4$ | $(1,2,8,7)(3,6)(4,5)$ | $0$ |
$12$ | $4$ | $(1,5,8,4)$ | $0$ |
$32$ | $6$ | $(1,6,7,8,3,2)(4,5)$ | $1$ |
$24$ | $8$ | $(1,3,7,4,8,6,2,5)$ | $0$ |
$24$ | $8$ | $(1,2,5,6,8,7,4,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.