Basic invariants
| Dimension: | $4$ |
| Group: | $Q_8:S_4$ |
| Conductor: | \(68121\)\(\medspace = 3^{4} \cdot 29^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.0.160016229.1 |
| 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 8.4.3902635809081.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 3x^{6} - 2x^{5} + x^{4} + 3x^{3} - x^{2} - x + 3 \)
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The roots of $f$ are computed in an extension of $\Q_{ 79 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 79 }$:
\( x^{2} + 78x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 54 + 79 + 30\cdot 79^{2} + 48\cdot 79^{3} + 51\cdot 79^{4} + 73\cdot 79^{5} + 55\cdot 79^{6} + 50\cdot 79^{7} + 38\cdot 79^{8} + 25\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 58 + 72\cdot 79 + 14\cdot 79^{2} + 9\cdot 79^{4} + 30\cdot 79^{5} + 69\cdot 79^{6} + 69\cdot 79^{7} + 70\cdot 79^{8} + 53\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 47 a + 25 + \left(22 a + 38\right)\cdot 79 + \left(63 a + 7\right)\cdot 79^{2} + \left(36 a + 22\right)\cdot 79^{3} + \left(50 a + 76\right)\cdot 79^{4} + \left(74 a + 8\right)\cdot 79^{5} + \left(37 a + 73\right)\cdot 79^{6} + \left(7 a + 11\right)\cdot 79^{7} + \left(40 a + 4\right)\cdot 79^{8} + \left(32 a + 16\right)\cdot 79^{9} +O(79^{10})\)
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| $r_{ 4 }$ | $=$ |
\( 77 a + 6 + \left(11 a + 19\right)\cdot 79 + \left(35 a + 1\right)\cdot 79^{2} + \left(8 a + 61\right)\cdot 79^{3} + \left(14 a + 40\right)\cdot 79^{4} + \left(15 a + 24\right)\cdot 79^{5} + \left(44 a + 47\right)\cdot 79^{6} + \left(74 a + 53\right)\cdot 79^{7} + \left(57 a + 75\right)\cdot 79^{8} + \left(68 a + 10\right)\cdot 79^{9} +O(79^{10})\)
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| $r_{ 5 }$ | $=$ |
\( 2 a + 4 + \left(67 a + 33\right)\cdot 79 + \left(43 a + 24\right)\cdot 79^{2} + \left(70 a + 34\right)\cdot 79^{3} + \left(64 a + 46\right)\cdot 79^{4} + \left(63 a + 25\right)\cdot 79^{5} + \left(34 a + 76\right)\cdot 79^{6} + \left(4 a + 4\right)\cdot 79^{7} + \left(21 a + 59\right)\cdot 79^{8} + \left(10 a + 21\right)\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 64 a + 57 + \left(37 a + 2\right)\cdot 79 + \left(75 a + 76\right)\cdot 79^{2} + \left(35 a + 17\right)\cdot 79^{3} + \left(62 a + 27\right)\cdot 79^{4} + \left(65 a + 58\right)\cdot 79^{5} + \left(29 a + 75\right)\cdot 79^{6} + \left(41 a + 65\right)\cdot 79^{7} + \left(68 a + 1\right)\cdot 79^{8} + \left(17 a + 36\right)\cdot 79^{9} +O(79^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 32 a + 72 + \left(56 a + 13\right)\cdot 79 + \left(15 a + 48\right)\cdot 79^{2} + \left(42 a + 74\right)\cdot 79^{3} + \left(28 a + 10\right)\cdot 79^{4} + \left(4 a + 33\right)\cdot 79^{5} + \left(41 a + 36\right)\cdot 79^{6} + \left(71 a + 60\right)\cdot 79^{7} + \left(38 a + 36\right)\cdot 79^{8} + \left(46 a + 8\right)\cdot 79^{9} +O(79^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 15 a + 42 + \left(41 a + 55\right)\cdot 79 + \left(3 a + 34\right)\cdot 79^{2} + \left(43 a + 57\right)\cdot 79^{3} + \left(16 a + 53\right)\cdot 79^{4} + \left(13 a + 61\right)\cdot 79^{5} + \left(49 a + 39\right)\cdot 79^{6} + \left(37 a + 77\right)\cdot 79^{7} + \left(10 a + 28\right)\cdot 79^{8} + \left(61 a + 64\right)\cdot 79^{9} +O(79^{10})\)
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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 |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,2)(3,6)(4,5)(7,8)$ | $-4$ |
| $6$ | $2$ | $(1,2)(7,8)$ | $0$ |
| $12$ | $2$ | $(1,8)(2,7)(3,5)(4,6)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,7)(6,8)$ | $0$ |
| $32$ | $3$ | $(1,6,8)(2,3,7)$ | $1$ |
| $6$ | $4$ | $(1,3,2,6)(4,8,5,7)$ | $0$ |
| $6$ | $4$ | $(1,3,2,6)(4,7,5,8)$ | $0$ |
| $12$ | $4$ | $(1,7,2,8)$ | $2$ |
| $12$ | $4$ | $(1,2)(3,7,6,8)(4,5)$ | $-2$ |
| $32$ | $6$ | $(1,6,8,2,3,7)(4,5)$ | $-1$ |
| $24$ | $8$ | $(1,5,3,7,2,4,6,8)$ | $0$ |
| $24$ | $8$ | $(1,7,3,5,2,8,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.