Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(318916345856\)\(\medspace = 2^{14} \cdot 269^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 8.2.335111316544.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 16T1294 |
| Parity: | odd |
| Projective image: | $S_4\wr C_2$ |
| Projective field: | Galois closure of 8.2.335111316544.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 75 + 134\cdot 191 + 50\cdot 191^{2} + 140\cdot 191^{3} + 87\cdot 191^{4} + 97\cdot 191^{5} + 28\cdot 191^{6} + 67\cdot 191^{7} + 6\cdot 191^{8} + 70\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 35 a + 95 + \left(15 a + 32\right)\cdot 191 + \left(14 a + 13\right)\cdot 191^{2} + \left(36 a + 39\right)\cdot 191^{3} + \left(98 a + 162\right)\cdot 191^{4} + \left(117 a + 129\right)\cdot 191^{5} + \left(71 a + 176\right)\cdot 191^{6} + \left(156 a + 28\right)\cdot 191^{7} + \left(133 a + 67\right)\cdot 191^{8} + \left(189 a + 188\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 60 a + 98 + \left(28 a + 29\right)\cdot 191 + \left(49 a + 8\right)\cdot 191^{2} + \left(84 a + 19\right)\cdot 191^{3} + \left(51 a + 124\right)\cdot 191^{4} + \left(109 a + 44\right)\cdot 191^{5} + \left(175 a + 94\right)\cdot 191^{6} + \left(90 a + 137\right)\cdot 191^{7} + \left(173 a + 40\right)\cdot 191^{8} + \left(104 a + 139\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 116 a + 101 + \left(104 a + 87\right)\cdot 191 + \left(23 a + 117\right)\cdot 191^{2} + \left(a + 165\right)\cdot 191^{3} + \left(25 a + 166\right)\cdot 191^{4} + \left(59 a + 4\right)\cdot 191^{5} + \left(128 a + 29\right)\cdot 191^{6} + \left(95 a + 112\right)\cdot 191^{7} + \left(67 a + 27\right)\cdot 191^{8} + \left(190 a + 120\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 156 a + 130 + \left(175 a + 12\right)\cdot 191 + \left(176 a + 12\right)\cdot 191^{2} + \left(154 a + 61\right)\cdot 191^{3} + \left(92 a + 33\right)\cdot 191^{4} + \left(73 a + 149\right)\cdot 191^{5} + \left(119 a + 130\right)\cdot 191^{6} + \left(34 a + 113\right)\cdot 191^{7} + \left(57 a + 44\right)\cdot 191^{8} + \left(a + 53\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 83 + 11\cdot 191 + 115\cdot 191^{2} + 141\cdot 191^{3} + 98\cdot 191^{4} + 5\cdot 191^{5} + 46\cdot 191^{6} + 172\cdot 191^{7} + 72\cdot 191^{8} + 70\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 131 a + 158 + \left(162 a + 188\right)\cdot 191 + \left(141 a + 28\right)\cdot 191^{2} + \left(106 a + 54\right)\cdot 191^{3} + \left(139 a + 91\right)\cdot 191^{4} + \left(81 a + 102\right)\cdot 191^{5} + \left(15 a + 160\right)\cdot 191^{6} + \left(100 a + 52\right)\cdot 191^{7} + \left(17 a + 123\right)\cdot 191^{8} + \left(86 a + 70\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 75 a + 26 + \left(86 a + 76\right)\cdot 191 + \left(167 a + 36\right)\cdot 191^{2} + \left(189 a + 143\right)\cdot 191^{3} + \left(165 a + 190\right)\cdot 191^{4} + \left(131 a + 38\right)\cdot 191^{5} + \left(62 a + 98\right)\cdot 191^{6} + \left(95 a + 79\right)\cdot 191^{7} + \left(123 a + 190\right)\cdot 191^{8} + 51\cdot 191^{9} +O(191^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(3,7)(4,8)$ | $-3$ |
| $9$ | $2$ | $(1,5)(2,6)(3,7)(4,8)$ | $1$ |
| $12$ | $2$ | $(1,2)$ | $3$ |
| $24$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $3$ |
| $36$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $36$ | $2$ | $(1,2)(3,7)(4,8)$ | $-1$ |
| $16$ | $3$ | $(1,5,6)$ | $0$ |
| $64$ | $3$ | $(1,5,6)(4,7,8)$ | $0$ |
| $12$ | $4$ | $(3,4,7,8)$ | $-3$ |
| $36$ | $4$ | $(1,2,5,6)(3,4,7,8)$ | $1$ |
| $36$ | $4$ | $(1,2,5,6)(3,7)(4,8)$ | $1$ |
| $72$ | $4$ | $(1,3,5,7)(2,4,6,8)$ | $-1$ |
| $72$ | $4$ | $(1,2)(3,4,7,8)$ | $-1$ |
| $144$ | $4$ | $(1,4,2,3)(5,7)(6,8)$ | $1$ |
| $48$ | $6$ | $(1,6,5)(3,7)(4,8)$ | $0$ |
| $96$ | $6$ | $(1,2)(4,8,7)$ | $0$ |
| $192$ | $6$ | $(1,4,5,7,6,8)(2,3)$ | $0$ |
| $144$ | $8$ | $(1,3,2,4,5,7,6,8)$ | $-1$ |
| $96$ | $12$ | $(1,5,6)(3,4,7,8)$ | $0$ |