Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(564379103744\)\(\medspace = 2^{9} \cdot 1033^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.4.9109431471368.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | even |
| Determinant: | 1.8264.2t1.b.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.4.9109431471368.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{6} - 16x^{5} + 33x^{4} + 96x^{3} + 82x^{2} + 24x - 256 \)
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The roots of $f$ are computed in an extension of $\Q_{ 163 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 163 }$:
\( x^{2} + 159x + 2 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 40 + 107\cdot 163 + 163^{2} + 58\cdot 163^{3} + 72\cdot 163^{4} + 39\cdot 163^{5} + 47\cdot 163^{6} + 78\cdot 163^{7} + 95\cdot 163^{8} + 73\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 80 a + 128 + \left(153 a + 15\right)\cdot 163 + \left(118 a + 100\right)\cdot 163^{2} + \left(138 a + 145\right)\cdot 163^{3} + \left(124 a + 143\right)\cdot 163^{4} + \left(146 a + 95\right)\cdot 163^{5} + \left(150 a + 63\right)\cdot 163^{6} + \left(77 a + 86\right)\cdot 163^{7} + \left(102 a + 78\right)\cdot 163^{8} + \left(38 a + 92\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 132 a + 100 + \left(121 a + 28\right)\cdot 163 + \left(74 a + 139\right)\cdot 163^{2} + \left(59 a + 43\right)\cdot 163^{3} + \left(67 a + 141\right)\cdot 163^{4} + \left(83 a + 28\right)\cdot 163^{5} + \left(75 a + 6\right)\cdot 163^{6} + \left(115 a + 129\right)\cdot 163^{7} + \left(81 a + 138\right)\cdot 163^{8} + \left(157 a + 14\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 31 a + 139 + \left(41 a + 57\right)\cdot 163 + \left(88 a + 153\right)\cdot 163^{2} + \left(103 a + 43\right)\cdot 163^{3} + \left(95 a + 25\right)\cdot 163^{4} + \left(79 a + 132\right)\cdot 163^{5} + \left(87 a + 61\right)\cdot 163^{6} + \left(47 a + 26\right)\cdot 163^{7} + \left(81 a + 24\right)\cdot 163^{8} + \left(5 a + 74\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 a + 49 + \left(16 a + 95\right)\cdot 163 + \left(23 a + 80\right)\cdot 163^{2} + \left(55 a + 61\right)\cdot 163^{3} + \left(86 a + 156\right)\cdot 163^{4} + \left(86 a + 73\right)\cdot 163^{5} + \left(42 a + 107\right)\cdot 163^{6} + \left(19 a + 44\right)\cdot 163^{7} + \left(120 a + 74\right)\cdot 163^{8} + \left(137 a + 126\right)\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 123 a + 46 + \left(146 a + 120\right)\cdot 163 + \left(139 a + 156\right)\cdot 163^{2} + \left(107 a + 95\right)\cdot 163^{3} + \left(76 a + 120\right)\cdot 163^{4} + \left(76 a + 7\right)\cdot 163^{5} + \left(120 a + 28\right)\cdot 163^{6} + \left(143 a + 79\right)\cdot 163^{7} + \left(42 a + 46\right)\cdot 163^{8} + \left(25 a + 68\right)\cdot 163^{9} +O(163^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 28 + 3\cdot 163 + 87\cdot 163^{2} + 110\cdot 163^{3} + 139\cdot 163^{4} + 41\cdot 163^{5} + 143\cdot 163^{6} + 123\cdot 163^{7} + 109\cdot 163^{8} + 57\cdot 163^{9} +O(163^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 83 a + 122 + \left(9 a + 60\right)\cdot 163 + \left(44 a + 96\right)\cdot 163^{2} + \left(24 a + 92\right)\cdot 163^{3} + \left(38 a + 15\right)\cdot 163^{4} + \left(16 a + 69\right)\cdot 163^{5} + \left(12 a + 31\right)\cdot 163^{6} + \left(85 a + 84\right)\cdot 163^{7} + \left(60 a + 84\right)\cdot 163^{8} + \left(124 a + 144\right)\cdot 163^{9} +O(163^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,4)(3,8)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(5,7)$ | $-1$ |
| $16$ | $3$ | $(3,4,8)$ | $0$ |
| $64$ | $3$ | $(2,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(2,3,4,8)$ | $-3$ |
| $36$ | $4$ | $(1,5,6,7)(2,3,4,8)$ | $1$ |
| $36$ | $4$ | $(1,5,6,7)(2,4)(3,8)$ | $1$ |
| $72$ | $4$ | $(1,2,6,4)(3,7,8,5)$ | $-1$ |
| $72$ | $4$ | $(1,5,6,7)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,6)(3,8,4)(5,7)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,6,8,7)$ | $0$ |
| $144$ | $8$ | $(1,2,5,3,6,4,7,8)$ | $-1$ |
| $96$ | $12$ | $(1,5,6,7)(3,4,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.