Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(518400\)\(\medspace = 2^{8} \cdot 3^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.4478976000.1 |
| 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 8.0.6561000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} - 10x^{5} - 4x^{4} - 10x^{3} - 2x^{2} - 2x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 97 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 97 }$:
\( x^{2} + 96x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 + 22\cdot 97 + 37\cdot 97^{2} + 90\cdot 97^{3} + 28\cdot 97^{4} + 58\cdot 97^{5} + 53\cdot 97^{6} + 13\cdot 97^{7} + 28\cdot 97^{8} + 75\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 a + 10 + \left(42 a + 11\right)\cdot 97 + \left(66 a + 80\right)\cdot 97^{2} + \left(44 a + 73\right)\cdot 97^{3} + \left(67 a + 25\right)\cdot 97^{4} + \left(43 a + 41\right)\cdot 97^{5} + \left(24 a + 25\right)\cdot 97^{6} + \left(22 a + 37\right)\cdot 97^{7} + \left(14 a + 18\right)\cdot 97^{8} + \left(70 a + 33\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 a + 49 + \left(60 a + 80\right)\cdot 97 + \left(21 a + 19\right)\cdot 97^{2} + \left(70 a + 56\right)\cdot 97^{3} + \left(3 a + 11\right)\cdot 97^{4} + \left(93 a + 69\right)\cdot 97^{5} + \left(36 a + 86\right)\cdot 97^{6} + \left(75 a + 49\right)\cdot 97^{7} + \left(2 a + 29\right)\cdot 97^{8} + \left(55 a + 33\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 49 a + 58 + \left(54 a + 5\right)\cdot 97 + \left(30 a + 7\right)\cdot 97^{2} + \left(52 a + 52\right)\cdot 97^{3} + \left(29 a + 48\right)\cdot 97^{4} + \left(53 a + 17\right)\cdot 97^{5} + \left(72 a + 6\right)\cdot 97^{6} + \left(74 a + 35\right)\cdot 97^{7} + \left(82 a + 10\right)\cdot 97^{8} + \left(26 a + 89\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 a + 19 + \left(80 a + 58\right)\cdot 97 + \left(21 a + 67\right)\cdot 97^{2} + \left(47 a + 76\right)\cdot 97^{3} + \left(39 a + 43\right)\cdot 97^{4} + \left(22 a + 65\right)\cdot 97^{5} + \left(12 a + 88\right)\cdot 97^{6} + 21\cdot 97^{7} + \left(67 a + 26\right)\cdot 97^{8} + \left(65 a + 75\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 60 a + 86 + \left(36 a + 6\right)\cdot 97 + \left(75 a + 78\right)\cdot 97^{2} + \left(26 a + 7\right)\cdot 97^{3} + \left(93 a + 42\right)\cdot 97^{4} + \left(3 a + 61\right)\cdot 97^{5} + \left(60 a + 30\right)\cdot 97^{6} + \left(21 a + 88\right)\cdot 97^{7} + \left(94 a + 53\right)\cdot 97^{8} + \left(41 a + 85\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 64 a + 52 + \left(16 a + 8\right)\cdot 97 + \left(75 a + 9\right)\cdot 97^{2} + \left(49 a + 5\right)\cdot 97^{3} + \left(57 a + 36\right)\cdot 97^{4} + \left(74 a + 48\right)\cdot 97^{5} + \left(84 a + 78\right)\cdot 97^{6} + \left(96 a + 9\right)\cdot 97^{7} + \left(29 a + 93\right)\cdot 97^{8} + \left(31 a + 73\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 87 + 89\cdot 97^{2} + 25\cdot 97^{3} + 54\cdot 97^{4} + 26\cdot 97^{5} + 18\cdot 97^{6} + 35\cdot 97^{7} + 31\cdot 97^{8} + 19\cdot 97^{9} +O(97^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $6$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $12$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $24$ | $2$ | $(1,8)(2,6)(3,7)$ | $0$ |
| $32$ | $3$ | $(1,4,6)(3,8,5)$ | $1$ |
| $6$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $6$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ |
| $12$ | $4$ | $(2,3,7,6)$ | $2$ |
| $12$ | $4$ | $(1,8)(2,6,7,3)(4,5)$ | $-2$ |
| $32$ | $6$ | $(1,4,3,8,5,6)(2,7)$ | $-1$ |
| $24$ | $8$ | $(1,2,5,3,8,7,4,6)$ | $0$ |
| $24$ | $8$ | $(1,5,3,7,8,4,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.