Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(129600\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.1119744000.4 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 4x^{7} + 10x^{6} - 16x^{5} + 13x^{4} - 4x^{3} - 16x^{2} + 16x - 4 \)
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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:
| $r_{ 1 }$ | $=$ |
\( 73 a + 95 + \left(13 a + 72\right)\cdot 97 + \left(93 a + 76\right)\cdot 97^{2} + \left(94 a + 96\right)\cdot 97^{3} + \left(49 a + 77\right)\cdot 97^{4} + \left(60 a + 48\right)\cdot 97^{5} + \left(26 a + 73\right)\cdot 97^{6} + \left(55 a + 5\right)\cdot 97^{7} + \left(68 a + 28\right)\cdot 97^{8} + \left(96 a + 27\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 85\cdot 97 + 45\cdot 97^{2} + 18\cdot 97^{3} + 39\cdot 97^{4} + 58\cdot 97^{5} + 51\cdot 97^{6} + 94\cdot 97^{7} + 61\cdot 97^{8} + 2\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 77 + 11\cdot 97 + 51\cdot 97^{2} + 78\cdot 97^{3} + 57\cdot 97^{4} + 38\cdot 97^{5} + 45\cdot 97^{6} + 2\cdot 97^{7} + 35\cdot 97^{8} + 94\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 a + 71 + \left(83 a + 13\right)\cdot 97 + \left(3 a + 59\right)\cdot 97^{2} + \left(2 a + 1\right)\cdot 97^{3} + \left(47 a + 33\right)\cdot 97^{4} + \left(36 a + 59\right)\cdot 97^{5} + \left(70 a + 39\right)\cdot 97^{6} + \left(41 a + 34\right)\cdot 97^{7} + \left(28 a + 41\right)\cdot 97^{8} + 55\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 73 a + 27 + \left(13 a + 83\right)\cdot 97 + \left(93 a + 37\right)\cdot 97^{2} + \left(94 a + 95\right)\cdot 97^{3} + \left(49 a + 63\right)\cdot 97^{4} + \left(60 a + 37\right)\cdot 97^{5} + \left(26 a + 57\right)\cdot 97^{6} + \left(55 a + 62\right)\cdot 97^{7} + \left(68 a + 55\right)\cdot 97^{8} + \left(96 a + 41\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 24 a + 3 + \left(83 a + 24\right)\cdot 97 + \left(3 a + 20\right)\cdot 97^{2} + 2 a\cdot 97^{3} + \left(47 a + 19\right)\cdot 97^{4} + \left(36 a + 48\right)\cdot 97^{5} + \left(70 a + 23\right)\cdot 97^{6} + \left(41 a + 91\right)\cdot 97^{7} + \left(28 a + 68\right)\cdot 97^{8} + 69\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 81 a + 57 + \left(87 a + 93\right)\cdot 97 + \left(39 a + 23\right)\cdot 97^{2} + \left(88 a + 24\right)\cdot 97^{3} + \left(47 a + 20\right)\cdot 97^{4} + 47 a\cdot 97^{5} + \left(56 a + 44\right)\cdot 97^{6} + \left(82 a + 35\right)\cdot 97^{7} + \left(94 a + 42\right)\cdot 97^{8} + \left(32 a + 79\right)\cdot 97^{9} +O(97^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 16 a + 41 + \left(9 a + 3\right)\cdot 97 + \left(57 a + 73\right)\cdot 97^{2} + \left(8 a + 72\right)\cdot 97^{3} + \left(49 a + 76\right)\cdot 97^{4} + \left(49 a + 96\right)\cdot 97^{5} + \left(40 a + 52\right)\cdot 97^{6} + \left(14 a + 61\right)\cdot 97^{7} + \left(2 a + 54\right)\cdot 97^{8} + \left(64 a + 17\right)\cdot 97^{9} +O(97^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,6)(2,3)(4,5)(7,8)$ | $-4$ |
| $6$ | $2$ | $(2,3)(4,5)$ | $0$ |
| $12$ | $2$ | $(1,2)(3,6)(4,8)(5,7)$ | $0$ |
| $24$ | $2$ | $(1,6)(2,5)(3,4)$ | $0$ |
| $32$ | $3$ | $(1,5,8)(4,7,6)$ | $1$ |
| $6$ | $4$ | $(1,2,6,3)(4,8,5,7)$ | $0$ |
| $6$ | $4$ | $(1,4,6,5)(2,8,3,7)$ | $0$ |
| $12$ | $4$ | $(1,6)(2,4,3,5)(7,8)$ | $-2$ |
| $12$ | $4$ | $(2,5,3,4)$ | $2$ |
| $32$ | $6$ | $(1,7,5,6,8,4)(2,3)$ | $-1$ |
| $24$ | $8$ | $(1,8,4,3,6,7,5,2)$ | $0$ |
| $24$ | $8$ | $(1,5,2,7,6,4,3,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.