Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(350464\)\(\medspace = 2^{8} \cdot 37^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.207474688.1 |
| 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 8.4.656825960704.2 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} - 8x^{5} + 12x^{4} - 10x^{3} + 12x^{2} - 4x + 3 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 a + 13 + \left(8 a + 13\right)\cdot 31 + \left(25 a + 29\right)\cdot 31^{2} + \left(21 a + 19\right)\cdot 31^{3} + \left(17 a + 4\right)\cdot 31^{4} + \left(27 a + 25\right)\cdot 31^{5} + \left(5 a + 12\right)\cdot 31^{6} + \left(2 a + 5\right)\cdot 31^{7} + \left(4 a + 13\right)\cdot 31^{8} + \left(22 a + 18\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 26 a + 8 + \left(4 a + 23\right)\cdot 31 + \left(4 a + 19\right)\cdot 31^{2} + \left(23 a + 26\right)\cdot 31^{3} + \left(19 a + 25\right)\cdot 31^{4} + 30\cdot 31^{5} + \left(27 a + 27\right)\cdot 31^{6} + \left(13 a + 8\right)\cdot 31^{7} + 21 a\cdot 31^{8} + \left(24 a + 29\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 8\cdot 31 + 9\cdot 31^{2} + 29\cdot 31^{3} + 29\cdot 31^{4} + 11\cdot 31^{5} + 18\cdot 31^{6} + 7\cdot 31^{7} + 8\cdot 31^{8} + 22\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 29 a + 19 + \left(20 a + 7\right)\cdot 31 + \left(13 a + 14\right)\cdot 31^{2} + \left(19 a + 23\right)\cdot 31^{3} + \left(14 a + 28\right)\cdot 31^{4} + \left(9 a + 25\right)\cdot 31^{5} + \left(23 a + 28\right)\cdot 31^{6} + \left(22 a + 24\right)\cdot 31^{7} + \left(30 a + 6\right)\cdot 31^{8} + \left(3 a + 26\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 5 a + 29 + \left(26 a + 6\right)\cdot 31 + \left(26 a + 23\right)\cdot 31^{2} + \left(7 a + 6\right)\cdot 31^{3} + \left(11 a + 11\right)\cdot 31^{4} + \left(30 a + 12\right)\cdot 31^{5} + \left(3 a + 19\right)\cdot 31^{6} + \left(17 a + 9\right)\cdot 31^{7} + \left(9 a + 29\right)\cdot 31^{8} + \left(6 a + 25\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 18 a + 8 + \left(22 a + 17\right)\cdot 31 + \left(5 a + 9\right)\cdot 31^{2} + \left(9 a + 7\right)\cdot 31^{3} + \left(13 a + 18\right)\cdot 31^{4} + 3 a\cdot 31^{5} + \left(25 a + 28\right)\cdot 31^{6} + \left(28 a + 3\right)\cdot 31^{7} + \left(26 a + 19\right)\cdot 31^{8} + \left(8 a + 27\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 2 a + 15 + \left(10 a + 20\right)\cdot 31 + \left(17 a + 20\right)\cdot 31^{2} + \left(11 a + 17\right)\cdot 31^{3} + \left(16 a + 7\right)\cdot 31^{4} + \left(21 a + 30\right)\cdot 31^{5} + \left(7 a + 3\right)\cdot 31^{6} + \left(8 a + 16\right)\cdot 31^{7} + 14\cdot 31^{8} + \left(27 a + 3\right)\cdot 31^{9} +O(31^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 18 + 26\cdot 31 + 28\cdot 31^{2} + 23\cdot 31^{3} + 28\cdot 31^{4} + 17\cdot 31^{5} + 15\cdot 31^{6} + 16\cdot 31^{7} + 31^{8} + 2\cdot 31^{9} +O(31^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $4$ | |
| $1$ | $2$ | $(1,4)(2,5)(3,8)(6,7)$ | $-4$ | |
| $6$ | $2$ | $(3,8)(6,7)$ | $0$ | |
| $12$ | $2$ | $(1,8)(2,6)(3,4)(5,7)$ | $0$ | ✓ |
| $24$ | $2$ | $(1,7)(2,5)(4,6)$ | $0$ | |
| $32$ | $3$ | $(2,6,8)(3,5,7)$ | $1$ | |
| $6$ | $4$ | $(1,6,4,7)(2,8,5,3)$ | $0$ | |
| $6$ | $4$ | $(1,5,4,2)(3,7,8,6)$ | $0$ | |
| $12$ | $4$ | $(1,6,4,7)(2,5)(3,8)$ | $-2$ | |
| $12$ | $4$ | $(1,8,4,3)$ | $2$ | |
| $32$ | $6$ | $(1,4)(2,3,6,5,8,7)$ | $-1$ | |
| $24$ | $8$ | $(1,5,6,3,4,2,7,8)$ | $0$ | |
| $24$ | $8$ | $(1,8,5,6,4,3,2,7)$ | $0$ |