Basic invariants
Dimension: | $4$ |
Group: | $Q_8:S_4$ |
Conductor: | \(118336\)\(\medspace = 2^{6} \cdot 43^{2} \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 8.2.81415168.1 |
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 8.0.1618268940544.5 |
Defining polynomial
$f(x)$ | $=$ | \( x^{8} - 4x^{7} + 8x^{6} - 8x^{5} + 2x^{4} + 6x^{3} - 12x^{2} + 6x - 1 \) . |
The roots of $f$ are computed in an extension of $\Q_{ 37 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 37 }$: \( x^{2} + 33x + 2 \)
Roots:
$r_{ 1 }$ | $=$ | \( 8 a + 33 + \left(5 a + 26\right)\cdot 37 + \left(21 a + 33\right)\cdot 37^{2} + 19\cdot 37^{3} + \left(16 a + 2\right)\cdot 37^{4} + \left(35 a + 1\right)\cdot 37^{5} + \left(5 a + 9\right)\cdot 37^{6} + \left(33 a + 13\right)\cdot 37^{7} + \left(3 a + 22\right)\cdot 37^{8} + \left(21 a + 6\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 2 }$ | $=$ | \( 29 a + 28 + \left(31 a + 2\right)\cdot 37 + \left(15 a + 2\right)\cdot 37^{2} + \left(36 a + 1\right)\cdot 37^{3} + \left(20 a + 29\right)\cdot 37^{4} + \left(a + 15\right)\cdot 37^{5} + \left(31 a + 34\right)\cdot 37^{6} + \left(3 a + 28\right)\cdot 37^{7} + \left(33 a + 4\right)\cdot 37^{8} + \left(15 a + 13\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 3 }$ | $=$ | \( 32 + 17\cdot 37 + 30\cdot 37^{2} + 21\cdot 37^{3} + 3\cdot 37^{4} + 36\cdot 37^{6} + 31\cdot 37^{7} + 9\cdot 37^{8} + 19\cdot 37^{9} +O(37^{10})\) |
$r_{ 4 }$ | $=$ | \( 31 a + 3 + \left(34 a + 3\right)\cdot 37 + \left(a + 3\right)\cdot 37^{2} + \left(9 a + 4\right)\cdot 37^{3} + \left(33 a + 21\right)\cdot 37^{4} + \left(13 a + 30\right)\cdot 37^{5} + \left(3 a + 30\right)\cdot 37^{6} + \left(5 a + 5\right)\cdot 37^{7} + \left(32 a + 19\right)\cdot 37^{8} + \left(9 a + 12\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 5 }$ | $=$ | \( 6 a + 16 + 2 a\cdot 37 + \left(35 a + 13\right)\cdot 37^{2} + \left(27 a + 1\right)\cdot 37^{3} + \left(3 a + 34\right)\cdot 37^{4} + \left(23 a + 15\right)\cdot 37^{5} + \left(33 a + 30\right)\cdot 37^{6} + \left(31 a + 22\right)\cdot 37^{7} + \left(4 a + 31\right)\cdot 37^{8} + \left(27 a + 19\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 6 }$ | $=$ | \( 9 + 12\cdot 37 + 28\cdot 37^{2} + 14\cdot 37^{3} + 14\cdot 37^{4} + 21\cdot 37^{5} + 33\cdot 37^{6} + 19\cdot 37^{7} + 25\cdot 37^{8} + 9\cdot 37^{9} +O(37^{10})\) |
$r_{ 7 }$ | $=$ | \( 16 a + 2 + \left(19 a + 30\right)\cdot 37 + \left(a + 6\right)\cdot 37^{2} + \left(23 a + 34\right)\cdot 37^{3} + \left(17 a + 34\right)\cdot 37^{4} + \left(7 a + 6\right)\cdot 37^{5} + \left(14 a + 36\right)\cdot 37^{6} + \left(16 a + 23\right)\cdot 37^{7} + \left(9 a + 6\right)\cdot 37^{8} + \left(16 a + 24\right)\cdot 37^{9} +O(37^{10})\) |
$r_{ 8 }$ | $=$ | \( 21 a + 29 + \left(17 a + 17\right)\cdot 37 + \left(35 a + 30\right)\cdot 37^{2} + \left(13 a + 13\right)\cdot 37^{3} + \left(19 a + 8\right)\cdot 37^{4} + \left(29 a + 19\right)\cdot 37^{5} + \left(22 a + 11\right)\cdot 37^{6} + \left(20 a + 1\right)\cdot 37^{7} + \left(27 a + 28\right)\cdot 37^{8} + \left(20 a + 5\right)\cdot 37^{9} +O(37^{10})\) |
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 |
$1$ | $1$ | $()$ | $4$ |
$1$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $-4$ |
$6$ | $2$ | $(4,8)(5,7)$ | $0$ |
$12$ | $2$ | $(1,7)(2,5)(3,8)(4,6)$ | $0$ |
$24$ | $2$ | $(1,4)(2,8)(5,7)$ | $0$ |
$32$ | $3$ | $(1,7,6)(2,5,3)$ | $1$ |
$6$ | $4$ | $(1,6,2,3)(4,5,8,7)$ | $0$ |
$6$ | $4$ | $(1,7,2,5)(3,8,6,4)$ | $0$ |
$12$ | $4$ | $(1,4,2,8)(3,6)(5,7)$ | $-2$ |
$12$ | $4$ | $(1,4,2,8)$ | $2$ |
$32$ | $6$ | $(1,5,6,2,7,3)(4,8)$ | $-1$ |
$24$ | $8$ | $(1,4,6,5,2,8,3,7)$ | $0$ |
$24$ | $8$ | $(1,4,7,3,2,8,5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.