Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(515078148096\)\(\medspace = 2^{12} \cdot 3^{3} \cdot 167^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.221626368.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.2004.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.221626368.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} + 2x^{6} + 2x^{5} - 4x^{4} + 2x^{3} + 2x^{2} - 6x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{2} + 192x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 47 a + 165 + \left(136 a + 63\right)\cdot 193 + \left(122 a + 98\right)\cdot 193^{2} + \left(30 a + 165\right)\cdot 193^{3} + \left(48 a + 170\right)\cdot 193^{4} + \left(127 a + 156\right)\cdot 193^{5} + \left(68 a + 40\right)\cdot 193^{6} + \left(22 a + 161\right)\cdot 193^{7} + \left(106 a + 188\right)\cdot 193^{8} + \left(112 a + 3\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 146 a + 19 + \left(56 a + 153\right)\cdot 193 + \left(70 a + 84\right)\cdot 193^{2} + \left(162 a + 73\right)\cdot 193^{3} + \left(144 a + 188\right)\cdot 193^{4} + \left(65 a + 42\right)\cdot 193^{5} + \left(124 a + 175\right)\cdot 193^{6} + \left(170 a + 114\right)\cdot 193^{7} + \left(86 a + 79\right)\cdot 193^{8} + \left(80 a + 10\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 151 a + 73 + \left(164 a + 129\right)\cdot 193 + \left(29 a + 146\right)\cdot 193^{2} + \left(191 a + 46\right)\cdot 193^{3} + \left(50 a + 125\right)\cdot 193^{4} + \left(107 a + 110\right)\cdot 193^{5} + \left(95 a + 119\right)\cdot 193^{6} + \left(158 a + 41\right)\cdot 193^{7} + \left(38 a + 155\right)\cdot 193^{8} + \left(114 a + 50\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 91 a + 63 + \left(37 a + 142\right)\cdot 193 + \left(73 a + 59\right)\cdot 193^{2} + \left(39 a + 103\right)\cdot 193^{3} + \left(8 a + 44\right)\cdot 193^{4} + \left(131 a + 93\right)\cdot 193^{5} + \left(51 a + 103\right)\cdot 193^{6} + \left(73 a + 75\right)\cdot 193^{7} + \left(46 a + 94\right)\cdot 193^{8} + \left(108 a + 191\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 42 a + 31 + \left(28 a + 143\right)\cdot 193 + \left(163 a + 11\right)\cdot 193^{2} + \left(a + 15\right)\cdot 193^{3} + \left(142 a + 178\right)\cdot 193^{4} + \left(85 a + 166\right)\cdot 193^{5} + \left(97 a + 107\right)\cdot 193^{6} + \left(34 a + 104\right)\cdot 193^{7} + \left(154 a + 35\right)\cdot 193^{8} + \left(78 a + 126\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 150 + 167\cdot 193 + 16\cdot 193^{2} + 3\cdot 193^{3} + 43\cdot 193^{4} + 61\cdot 193^{5} + 162\cdot 193^{6} + 28\cdot 193^{7} + 92\cdot 193^{8} + 161\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 119 + 76\cdot 193 + 65\cdot 193^{2} + 102\cdot 193^{3} + 8\cdot 193^{4} + 117\cdot 193^{5} + 38\cdot 193^{6} + 148\cdot 193^{7} + 58\cdot 193^{8} + 167\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 102 a + 154 + \left(155 a + 88\right)\cdot 193 + \left(119 a + 95\right)\cdot 193^{2} + \left(153 a + 69\right)\cdot 193^{3} + \left(184 a + 13\right)\cdot 193^{4} + \left(61 a + 23\right)\cdot 193^{5} + \left(141 a + 24\right)\cdot 193^{6} + \left(119 a + 97\right)\cdot 193^{7} + \left(146 a + 67\right)\cdot 193^{8} + \left(84 a + 60\right)\cdot 193^{9} +O(193^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,6)(5,7)$ | $-3$ |
| $9$ | $2$ | $(1,4)(2,8)(3,6)(5,7)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,5)$ | $1$ |
| $36$ | $2$ | $(1,2)(3,6)(5,7)$ | $-1$ |
| $16$ | $3$ | $(1,4,8)$ | $0$ |
| $64$ | $3$ | $(1,4,8)(5,6,7)$ | $0$ |
| $12$ | $4$ | $(3,5,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,2,4,8)(3,5,6,7)$ | $1$ |
| $36$ | $4$ | $(1,2,4,8)(3,6)(5,7)$ | $1$ |
| $72$ | $4$ | $(1,3,4,6)(2,5,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,2)(3,5,6,7)$ | $-1$ |
| $144$ | $4$ | $(1,5,2,3)(4,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,8,4)(3,6)(5,7)$ | $0$ |
| $96$ | $6$ | $(1,2)(5,7,6)$ | $0$ |
| $192$ | $6$ | $(1,5,4,6,8,7)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,3,2,5,4,6,8,7)$ | $-1$ |
| $96$ | $12$ | $(1,4,8)(3,5,6,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.