Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(117785509888\)\(\medspace = 2^{14} \cdot 193^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.88799232064.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Determinant: | 1.772.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.88799232064.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} - 8x^{5} + 13x^{4} + 16x^{3} - 2x^{2} - 36x - 28 \)
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The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{2} + 193x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 38 a + 114 + \left(192 a + 193\right)\cdot 199 + \left(107 a + 6\right)\cdot 199^{2} + \left(22 a + 155\right)\cdot 199^{3} + \left(96 a + 30\right)\cdot 199^{4} + \left(64 a + 180\right)\cdot 199^{5} + \left(26 a + 86\right)\cdot 199^{6} + \left(87 a + 180\right)\cdot 199^{7} + \left(11 a + 166\right)\cdot 199^{8} + \left(71 a + 31\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 173 a + 16 + \left(65 a + 98\right)\cdot 199 + \left(128 a + 90\right)\cdot 199^{2} + \left(175 a + 185\right)\cdot 199^{3} + \left(132 a + 194\right)\cdot 199^{4} + \left(173 a + 191\right)\cdot 199^{5} + \left(191 a + 66\right)\cdot 199^{6} + \left(141 a + 182\right)\cdot 199^{7} + \left(61 a + 130\right)\cdot 199^{8} + \left(34 a + 154\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 26 a + 59 + \left(133 a + 121\right)\cdot 199 + \left(70 a + 197\right)\cdot 199^{2} + \left(23 a + 115\right)\cdot 199^{3} + \left(66 a + 20\right)\cdot 199^{4} + \left(25 a + 106\right)\cdot 199^{5} + \left(7 a + 49\right)\cdot 199^{6} + \left(57 a + 46\right)\cdot 199^{7} + \left(137 a + 160\right)\cdot 199^{8} + \left(164 a + 99\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 168 + 55\cdot 199 + 84\cdot 199^{2} + 135\cdot 199^{3} + 33\cdot 199^{4} + 47\cdot 199^{5} + 138\cdot 199^{6} + 38\cdot 199^{7} + 11\cdot 199^{8} + 102\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 86 a + 3 + \left(26 a + 53\right)\cdot 199 + \left(166 a + 67\right)\cdot 199^{2} + \left(141 a + 4\right)\cdot 199^{3} + \left(153 a + 99\right)\cdot 199^{4} + \left(162 a + 36\right)\cdot 199^{5} + \left(96 a + 31\right)\cdot 199^{6} + \left(150 a + 179\right)\cdot 199^{7} + \left(153 a + 164\right)\cdot 199^{8} + \left(108 a + 120\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 172 + 33\cdot 199 + 43\cdot 199^{2} + 124\cdot 199^{3} + 146\cdot 199^{4} + 97\cdot 199^{5} + 191\cdot 199^{6} + 98\cdot 199^{7} + 71\cdot 199^{8} + 16\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 161 a + 143 + \left(6 a + 114\right)\cdot 199 + \left(91 a + 64\right)\cdot 199^{2} + \left(176 a + 182\right)\cdot 199^{3} + \left(102 a + 186\right)\cdot 199^{4} + \left(134 a + 72\right)\cdot 199^{5} + \left(172 a + 180\right)\cdot 199^{6} + \left(111 a + 79\right)\cdot 199^{7} + \left(187 a + 148\right)\cdot 199^{8} + \left(127 a + 48\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 113 a + 121 + \left(172 a + 125\right)\cdot 199 + \left(32 a + 42\right)\cdot 199^{2} + \left(57 a + 92\right)\cdot 199^{3} + \left(45 a + 83\right)\cdot 199^{4} + \left(36 a + 63\right)\cdot 199^{5} + \left(102 a + 51\right)\cdot 199^{6} + \left(48 a + 189\right)\cdot 199^{7} + \left(45 a + 140\right)\cdot 199^{8} + \left(90 a + 22\right)\cdot 199^{9} +O(199^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(1,6)(4,7)$ | $-3$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $1$ |
| $12$ | $2$ | $(2,3)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $3$ |
| $36$ | $2$ | $(1,4)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,3)(4,7)$ | $-1$ |
| $16$ | $3$ | $(2,5,8)$ | $0$ |
| $64$ | $3$ | $(2,5,8)(4,6,7)$ | $0$ |
| $12$ | $4$ | $(1,4,6,7)$ | $-3$ |
| $36$ | $4$ | $(1,4,6,7)(2,3,5,8)$ | $1$ |
| $36$ | $4$ | $(1,6)(2,3,5,8)(4,7)$ | $1$ |
| $72$ | $4$ | $(1,5,6,2)(3,4,8,7)$ | $-1$ |
| $72$ | $4$ | $(1,4,6,7)(2,3)$ | $-1$ |
| $144$ | $4$ | $(1,2,4,3)(5,6)(7,8)$ | $1$ |
| $48$ | $6$ | $(1,6)(2,8,5)(4,7)$ | $0$ |
| $96$ | $6$ | $(2,3)(4,7,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,4,5,6,8,7)$ | $0$ |
| $144$ | $8$ | $(1,3,4,5,6,8,7,2)$ | $-1$ |
| $96$ | $12$ | $(1,4,6,7)(2,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.