Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(313600\)\(\medspace = 2^{8} \cdot 5^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.3073280000.1 |
| 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 8.4.6294077440000.46 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + x^{6} - 3x^{4} + 8 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 29 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 29 }$:
\( x^{2} + 24x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 a + 6 + \left(a + 14\right)\cdot 29 + \left(3 a + 19\right)\cdot 29^{2} + \left(28 a + 27\right)\cdot 29^{3} + \left(28 a + 2\right)\cdot 29^{4} + \left(2 a + 24\right)\cdot 29^{5} + \left(18 a + 8\right)\cdot 29^{6} + \left(27 a + 21\right)\cdot 29^{7} + \left(28 a + 2\right)\cdot 29^{8} + \left(25 a + 14\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 a + 27 + \left(27 a + 14\right)\cdot 29 + \left(17 a + 12\right)\cdot 29^{2} + \left(17 a + 8\right)\cdot 29^{3} + \left(10 a + 11\right)\cdot 29^{4} + \left(23 a + 19\right)\cdot 29^{5} + \left(4 a + 28\right)\cdot 29^{6} + \left(25 a + 11\right)\cdot 29^{7} + \left(17 a + 11\right)\cdot 29^{8} + \left(19 a + 3\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 + 3\cdot 29 + 29^{2} + 20\cdot 29^{3} + 13\cdot 29^{4} + 20\cdot 29^{5} + 2\cdot 29^{6} + 17\cdot 29^{7} + 24\cdot 29^{8} + 18\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 18 a + 3 + \left(27 a + 10\right)\cdot 29 + \left(25 a + 4\right)\cdot 29^{2} + 20\cdot 29^{3} + 3\cdot 29^{4} + \left(26 a + 10\right)\cdot 29^{5} + \left(10 a + 9\right)\cdot 29^{6} + \left(a + 25\right)\cdot 29^{7} + 3\cdot 29^{8} + \left(3 a + 28\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 a + 23 + \left(27 a + 14\right)\cdot 29 + \left(25 a + 9\right)\cdot 29^{2} + 29^{3} + 26\cdot 29^{4} + \left(26 a + 4\right)\cdot 29^{5} + \left(10 a + 20\right)\cdot 29^{6} + \left(a + 7\right)\cdot 29^{7} + 26\cdot 29^{8} + \left(3 a + 14\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 5 a + 2 + \left(a + 14\right)\cdot 29 + \left(11 a + 16\right)\cdot 29^{2} + \left(11 a + 20\right)\cdot 29^{3} + \left(18 a + 17\right)\cdot 29^{4} + \left(5 a + 9\right)\cdot 29^{5} + 24 a\cdot 29^{6} + \left(3 a + 17\right)\cdot 29^{7} + \left(11 a + 17\right)\cdot 29^{8} + \left(9 a + 25\right)\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 22 + 25\cdot 29 + 27\cdot 29^{2} + 8\cdot 29^{3} + 15\cdot 29^{4} + 8\cdot 29^{5} + 26\cdot 29^{6} + 11\cdot 29^{7} + 4\cdot 29^{8} + 10\cdot 29^{9} +O(29^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 11 a + 26 + \left(a + 18\right)\cdot 29 + \left(3 a + 24\right)\cdot 29^{2} + \left(28 a + 8\right)\cdot 29^{3} + \left(28 a + 25\right)\cdot 29^{4} + \left(2 a + 18\right)\cdot 29^{5} + \left(18 a + 19\right)\cdot 29^{6} + \left(27 a + 3\right)\cdot 29^{7} + \left(28 a + 25\right)\cdot 29^{8} + 25 a\cdot 29^{9} +O(29^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $-4$ |
| $6$ | $2$ | $(1,5)(2,6)$ | $0$ |
| $12$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $0$ |
| $24$ | $2$ | $(1,6)(2,5)(4,8)$ | $0$ |
| $32$ | $3$ | $(1,8,7)(3,5,4)$ | $1$ |
| $6$ | $4$ | $(1,7,5,3)(2,4,6,8)$ | $0$ |
| $6$ | $4$ | $(1,8,5,4)(2,3,6,7)$ | $0$ |
| $12$ | $4$ | $(1,6,5,2)$ | $2$ |
| $12$ | $4$ | $(1,4,5,8)(2,6)(3,7)$ | $-2$ |
| $32$ | $6$ | $(1,3,8,5,7,4)(2,6)$ | $-1$ |
| $24$ | $8$ | $(1,4,7,6,5,8,3,2)$ | $0$ |
| $24$ | $8$ | $(1,2,8,3,5,6,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.