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 8.2.26363260928.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 4x^{6} - 14x^{5} + 15x^{4} + 22x^{3} - 16x^{2} - 116x - 157 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 37 a + 2 + \left(34 a + 9\right)\cdot 43 + \left(27 a + 24\right)\cdot 43^{2} + \left(11 a + 8\right)\cdot 43^{3} + \left(37 a + 14\right)\cdot 43^{4} + \left(35 a + 35\right)\cdot 43^{5} + \left(33 a + 5\right)\cdot 43^{6} + \left(40 a + 37\right)\cdot 43^{7} + \left(7 a + 33\right)\cdot 43^{8} + \left(42 a + 5\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 a + 39 + \left(8 a + 6\right)\cdot 43 + \left(15 a + 17\right)\cdot 43^{2} + \left(31 a + 35\right)\cdot 43^{3} + \left(5 a + 39\right)\cdot 43^{4} + \left(7 a + 33\right)\cdot 43^{5} + \left(9 a + 3\right)\cdot 43^{6} + \left(2 a + 1\right)\cdot 43^{7} + \left(35 a + 1\right)\cdot 43^{8} + 40\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 20\cdot 43 + 34\cdot 43^{2} + 4\cdot 43^{3} + 16\cdot 43^{4} + 26\cdot 43^{5} + 35\cdot 43^{6} + 8\cdot 43^{7} + 12\cdot 43^{8} + 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 42 + 40\cdot 43 + 5\cdot 43^{2} + 22\cdot 43^{3} + 11\cdot 43^{4} + 23\cdot 43^{5} + 20\cdot 43^{6} + 17\cdot 43^{7} + 14\cdot 43^{8} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 4 a + 25 + \left(37 a + 33\right)\cdot 43 + \left(11 a + 29\right)\cdot 43^{2} + \left(36 a + 6\right)\cdot 43^{3} + 3\cdot 43^{4} + 3\cdot 43^{5} + \left(27 a + 42\right)\cdot 43^{6} + \left(2 a + 41\right)\cdot 43^{7} + \left(28 a + 27\right)\cdot 43^{8} + \left(38 a + 26\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 a + 12 + \left(42 a + 18\right)\cdot 43 + \left(6 a + 24\right)\cdot 43^{2} + \left(24 a + 1\right)\cdot 43^{3} + \left(16 a + 42\right)\cdot 43^{4} + \left(4 a + 29\right)\cdot 43^{5} + \left(19 a + 11\right)\cdot 43^{6} + \left(28 a + 19\right)\cdot 43^{7} + \left(30 a + 13\right)\cdot 43^{8} + \left(12 a + 39\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 a + 29 + \left(5 a + 23\right)\cdot 43 + \left(31 a + 4\right)\cdot 43^{2} + \left(6 a + 31\right)\cdot 43^{3} + \left(42 a + 10\right)\cdot 43^{4} + \left(42 a + 2\right)\cdot 43^{5} + \left(15 a + 26\right)\cdot 43^{6} + \left(40 a + 17\right)\cdot 43^{7} + \left(14 a + 10\right)\cdot 43^{8} + \left(4 a + 37\right)\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( a + 11 + 19\cdot 43 + \left(36 a + 31\right)\cdot 43^{2} + \left(18 a + 18\right)\cdot 43^{3} + \left(26 a + 34\right)\cdot 43^{4} + \left(38 a + 17\right)\cdot 43^{5} + \left(23 a + 26\right)\cdot 43^{6} + \left(14 a + 28\right)\cdot 43^{7} + \left(12 a + 15\right)\cdot 43^{8} + \left(30 a + 21\right)\cdot 43^{9} +O(43^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $1$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $-8$ |
| $6$ | $2$ | $(3,4)(5,6)$ | $0$ |
| $12$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,6)(4,5)$ | $0$ |
| $32$ | $3$ | $(1,8,4)(2,7,3)$ | $-1$ |
| $6$ | $4$ | $(1,5,2,6)(3,7,4,8)$ | $0$ |
| $6$ | $4$ | $(1,6,2,5)(3,7,4,8)$ | $0$ |
| $12$ | $4$ | $(1,2)(3,5,4,6)(7,8)$ | $0$ |
| $12$ | $4$ | $(1,8,2,7)$ | $0$ |
| $32$ | $6$ | $(1,3,8,2,4,7)(5,6)$ | $1$ |
| $24$ | $8$ | $(1,8,5,3,2,7,6,4)$ | $0$ |
| $24$ | $8$ | $(1,4,7,6,2,3,8,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.