Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(304704\)\(\medspace = 2^{6} \cdot 3^{2} \cdot 23^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.7737043968.3 |
| 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 8.0.52225046784.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 6x^{6} + 6x^{5} - 46x^{4} + 102x^{3} - 118x^{2} + 72x - 18 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 41 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 41 }$:
\( x^{2} + 38x + 6 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 21 a + 3 + \left(7 a + 32\right)\cdot 41 + \left(23 a + 17\right)\cdot 41^{2} + \left(3 a + 22\right)\cdot 41^{3} + \left(23 a + 17\right)\cdot 41^{4} + \left(18 a + 20\right)\cdot 41^{5} + \left(35 a + 1\right)\cdot 41^{6} + \left(40 a + 24\right)\cdot 41^{7} + 8 a\cdot 41^{8} + \left(28 a + 13\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 19 + 3\cdot 41 + 38\cdot 41^{2} + 22\cdot 41^{3} + 37\cdot 41^{4} + 19\cdot 41^{5} + 27\cdot 41^{6} + 33\cdot 41^{7} + 17\cdot 41^{8} + 35\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 a + 40 + \left(13 a + 34\right)\cdot 41 + \left(8 a + 2\right)\cdot 41^{2} + \left(18 a + 17\right)\cdot 41^{3} + \left(27 a + 16\right)\cdot 41^{4} + \left(25 a + 27\right)\cdot 41^{5} + \left(15 a + 33\right)\cdot 41^{6} + \left(4 a + 37\right)\cdot 41^{7} + 33\cdot 41^{8} + \left(19 a + 29\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 a + 35 + \left(27 a + 21\right)\cdot 41 + \left(32 a + 14\right)\cdot 41^{2} + \left(22 a + 22\right)\cdot 41^{3} + \left(13 a + 39\right)\cdot 41^{4} + \left(15 a + 35\right)\cdot 41^{5} + \left(25 a + 13\right)\cdot 41^{6} + \left(36 a + 35\right)\cdot 41^{7} + \left(40 a + 29\right)\cdot 41^{8} + \left(21 a + 4\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 15 a + 33 + \left(5 a + 27\right)\cdot 41 + \left(3 a + 9\right)\cdot 41^{2} + \left(4 a + 3\right)\cdot 41^{3} + \left(27 a + 20\right)\cdot 41^{4} + \left(3 a + 35\right)\cdot 41^{5} + \left(9 a + 23\right)\cdot 41^{6} + \left(36 a + 32\right)\cdot 41^{7} + \left(23 a + 21\right)\cdot 41^{8} + \left(38 a + 36\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 26 a + 37 + \left(35 a + 28\right)\cdot 41 + \left(37 a + 13\right)\cdot 41^{2} + \left(36 a + 12\right)\cdot 41^{3} + \left(13 a + 15\right)\cdot 41^{4} + \left(37 a + 19\right)\cdot 41^{5} + \left(31 a + 6\right)\cdot 41^{6} + \left(4 a + 9\right)\cdot 41^{7} + \left(17 a + 16\right)\cdot 41^{8} + \left(2 a + 5\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 20 a + 25 + \left(33 a + 33\right)\cdot 41 + \left(17 a + 38\right)\cdot 41^{2} + \left(37 a + 9\right)\cdot 41^{3} + \left(17 a + 1\right)\cdot 41^{4} + \left(22 a + 12\right)\cdot 41^{5} + \left(5 a + 7\right)\cdot 41^{6} + 29\cdot 41^{7} + \left(32 a + 27\right)\cdot 41^{8} + \left(12 a + 6\right)\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 17 + 22\cdot 41 + 28\cdot 41^{2} + 12\cdot 41^{3} + 16\cdot 41^{4} + 34\cdot 41^{5} + 8\cdot 41^{6} + 3\cdot 41^{7} + 16\cdot 41^{8} + 32\cdot 41^{9} +O(41^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,8)(4,7)(5,6)$ | $-4$ |
| $6$ | $2$ | $(1,3)(5,6)$ | $0$ |
| $12$ | $2$ | $(1,2)(3,8)(4,6)(5,7)$ | $0$ |
| $24$ | $2$ | $(1,5)(3,6)(4,7)$ | $0$ |
| $32$ | $3$ | $(1,2,4)(3,8,7)$ | $1$ |
| $6$ | $4$ | $(1,6,3,5)(2,7,8,4)$ | $0$ |
| $6$ | $4$ | $(1,7,3,4)(2,6,8,5)$ | $0$ |
| $12$ | $4$ | $(1,8,3,2)(4,7)(5,6)$ | $-2$ |
| $12$ | $4$ | $(2,5,8,6)$ | $2$ |
| $32$ | $6$ | $(1,6,7,3,5,4)(2,8)$ | $-1$ |
| $24$ | $8$ | $(1,8,6,4,3,2,5,7)$ | $0$ |
| $24$ | $8$ | $(1,5,7,2,3,6,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.