Basic invariants
| Dimension: | $9$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(123\!\cdots\!416\)\(\medspace = 2^{16} \cdot 3^{11} \cdot 13^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.19680620544.5 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T274 |
| Parity: | odd |
| Determinant: | 1.39.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.19680620544.5 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 2x^{7} - 2x^{6} + 4x^{5} + 4x^{4} + 4x^{3} - 8x^{2} - 32x + 4 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$:
\( x^{2} + 190x + 19 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 170 a + 152 + \left(179 a + 79\right)\cdot 191 + \left(132 a + 144\right)\cdot 191^{2} + \left(129 a + 2\right)\cdot 191^{3} + \left(121 a + 171\right)\cdot 191^{4} + \left(93 a + 28\right)\cdot 191^{5} + \left(156 a + 116\right)\cdot 191^{6} + \left(100 a + 45\right)\cdot 191^{7} + \left(113 a + 142\right)\cdot 191^{8} + \left(72 a + 148\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 134 a + 126 + \left(89 a + 189\right)\cdot 191 + \left(169 a + 153\right)\cdot 191^{2} + \left(19 a + 53\right)\cdot 191^{3} + \left(11 a + 129\right)\cdot 191^{4} + \left(68 a + 6\right)\cdot 191^{5} + \left(23 a + 4\right)\cdot 191^{6} + \left(39 a + 73\right)\cdot 191^{7} + \left(167 a + 90\right)\cdot 191^{8} + \left(45 a + 2\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 57 a + 69 + \left(101 a + 145\right)\cdot 191 + \left(21 a + 42\right)\cdot 191^{2} + \left(171 a + 95\right)\cdot 191^{3} + \left(179 a + 120\right)\cdot 191^{4} + \left(122 a + 63\right)\cdot 191^{5} + \left(167 a + 150\right)\cdot 191^{6} + \left(151 a + 88\right)\cdot 191^{7} + \left(23 a + 27\right)\cdot 191^{8} + \left(145 a + 72\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 189 a + 83 + \left(124 a + 61\right)\cdot 191 + \left(160 a + 70\right)\cdot 191^{2} + \left(132 a + 86\right)\cdot 191^{3} + \left(50 a + 2\right)\cdot 191^{4} + \left(148 a + 107\right)\cdot 191^{5} + \left(77 a + 157\right)\cdot 191^{6} + \left(81 a + 173\right)\cdot 191^{7} + \left(160 a + 85\right)\cdot 191^{8} + \left(48 a + 146\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 2 a + 81 + \left(66 a + 188\right)\cdot 191 + \left(30 a + 105\right)\cdot 191^{2} + \left(58 a + 58\right)\cdot 191^{3} + \left(140 a + 111\right)\cdot 191^{4} + \left(42 a + 13\right)\cdot 191^{5} + \left(113 a + 87\right)\cdot 191^{6} + \left(109 a + 177\right)\cdot 191^{7} + \left(30 a + 164\right)\cdot 191^{8} + \left(142 a + 34\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 148 + 19\cdot 191 + 69\cdot 191^{2} + 44\cdot 191^{3} + 178\cdot 191^{4} + 170\cdot 191^{5} + 105\cdot 191^{6} + 8\cdot 191^{7} + 162\cdot 191^{8} + 90\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 21 a + 131 + \left(11 a + 89\right)\cdot 191 + \left(58 a + 97\right)\cdot 191^{2} + \left(61 a + 190\right)\cdot 191^{3} + \left(69 a + 162\right)\cdot 191^{4} + 97 a\cdot 191^{5} + \left(34 a + 179\right)\cdot 191^{6} + \left(90 a + 180\right)\cdot 191^{7} + \left(77 a + 154\right)\cdot 191^{8} + \left(118 a + 107\right)\cdot 191^{9} +O(191^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 167 + 180\cdot 191 + 79\cdot 191^{2} + 41\cdot 191^{3} + 79\cdot 191^{4} + 181\cdot 191^{5} + 154\cdot 191^{6} + 15\cdot 191^{7} + 127\cdot 191^{8} + 160\cdot 191^{9} +O(191^{10})\)
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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 |
| $1$ | $1$ | $()$ | $9$ |
| $6$ | $2$ | $(2,4)(3,5)$ | $-3$ |
| $9$ | $2$ | $(1,7)(2,4)(3,5)(6,8)$ | $1$ |
| $12$ | $2$ | $(1,6)$ | $3$ |
| $24$ | $2$ | $(1,2)(3,6)(4,7)(5,8)$ | $-3$ |
| $36$ | $2$ | $(1,6)(2,3)$ | $1$ |
| $36$ | $2$ | $(1,6)(2,4)(3,5)$ | $-1$ |
| $16$ | $3$ | $(1,7,8)$ | $0$ |
| $64$ | $3$ | $(1,7,8)(3,4,5)$ | $0$ |
| $12$ | $4$ | $(2,3,4,5)$ | $-3$ |
| $36$ | $4$ | $(1,6,7,8)(2,3,4,5)$ | $1$ |
| $36$ | $4$ | $(1,6,7,8)(2,4)(3,5)$ | $1$ |
| $72$ | $4$ | $(1,2,7,4)(3,8,5,6)$ | $1$ |
| $72$ | $4$ | $(1,6)(2,3,4,5)$ | $-1$ |
| $144$ | $4$ | $(1,3,6,2)(4,7)(5,8)$ | $-1$ |
| $48$ | $6$ | $(1,8,7)(2,4)(3,5)$ | $0$ |
| $96$ | $6$ | $(1,6)(3,5,4)$ | $0$ |
| $192$ | $6$ | $(1,3,7,4,8,5)(2,6)$ | $0$ |
| $144$ | $8$ | $(1,2,6,3,7,4,8,5)$ | $1$ |
| $96$ | $12$ | $(1,7,8)(2,3,4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.