Basic invariants
| Dimension: | $12$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(46194649946827075584\)\(\medspace = 2^{12} \cdot 3^{14} \cdot 11^{9} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.67625959104.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1763 |
| Parity: | odd |
| Determinant: | 1.11.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.67625959104.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} + 9x^{6} - 11x^{5} - 21x^{4} + 69x^{3} + 10x^{2} - 48x - 24 \)
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The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$:
\( x^{3} + 6x + 179 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 28 + 129\cdot 181 + 22\cdot 181^{2} + 171\cdot 181^{3} + 58\cdot 181^{4} + 93\cdot 181^{5} + 23\cdot 181^{6} + 153\cdot 181^{7} + 106\cdot 181^{8} + 157\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 129 + 2\cdot 181 + 142\cdot 181^{2} + 93\cdot 181^{3} + 152\cdot 181^{4} + 13\cdot 181^{5} + 33\cdot 181^{6} + 37\cdot 181^{7} + 6\cdot 181^{8} + 146\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 a^{2} + 146 a + 1 + \left(171 a^{2} + 4 a + 2\right)\cdot 181 + \left(175 a^{2} + 110 a + 48\right)\cdot 181^{2} + \left(50 a^{2} + 113 a + 7\right)\cdot 181^{3} + \left(3 a^{2} + 79 a + 7\right)\cdot 181^{4} + \left(92 a^{2} + 68 a + 50\right)\cdot 181^{5} + \left(70 a^{2} + 44 a + 84\right)\cdot 181^{6} + \left(29 a^{2} + 118 a + 159\right)\cdot 181^{7} + \left(23 a^{2} + 132 a + 166\right)\cdot 181^{8} + \left(33 a^{2} + 124 a + 164\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 41 a^{2} + 46 a + 27 + \left(50 a^{2} + 83 a + 115\right)\cdot 181 + \left(153 a^{2} + 41 a + 67\right)\cdot 181^{2} + \left(2 a^{2} + 143 a + 59\right)\cdot 181^{3} + \left(77 a^{2} + 36 a + 2\right)\cdot 181^{4} + \left(145 a^{2} + 16 a + 140\right)\cdot 181^{5} + \left(34 a^{2} + 76\right)\cdot 181^{6} + \left(132 a^{2} + 123 a + 61\right)\cdot 181^{7} + \left(71 a^{2} + 121 a + 54\right)\cdot 181^{8} + \left(51 a^{2} + 6 a + 132\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 60 a^{2} + 21 a + 103 + \left(24 a^{2} + 112 a + 11\right)\cdot 181 + \left(93 a^{2} + 61 a + 8\right)\cdot 181^{2} + \left(144 a^{2} + 113 a + 83\right)\cdot 181^{3} + \left(139 a^{2} + 40 a + 72\right)\cdot 181^{4} + \left(155 a^{2} + 136 a\right)\cdot 181^{5} + \left(86 a^{2} + 32 a + 104\right)\cdot 181^{6} + \left(2 a^{2} + 59 a + 85\right)\cdot 181^{7} + \left(94 a^{2} + 39 a + 143\right)\cdot 181^{8} + \left(63 a^{2} + 90 a + 180\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 66 a^{2} + 122 a + 48 + \left(140 a^{2} + 65 a + 60\right)\cdot 181 + \left(173 a^{2} + 61 a + 39\right)\cdot 181^{2} + \left(55 a^{2} + 114 a + 27\right)\cdot 181^{3} + \left(20 a^{2} + 77 a + 75\right)\cdot 181^{4} + \left(125 a^{2} + 23 a + 1\right)\cdot 181^{5} + \left(4 a^{2} + 13 a + 2\right)\cdot 181^{6} + \left(169 a^{2} + 144 a + 175\right)\cdot 181^{7} + \left(12 a^{2} + 170 a + 125\right)\cdot 181^{8} + \left(63 a^{2} + 60 a + 103\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 80 a^{2} + 114 a + 2 + \left(106 a^{2} + 166 a + 159\right)\cdot 181 + \left(115 a^{2} + 77 a + 97\right)\cdot 181^{2} + \left(33 a^{2} + 105 a + 1\right)\cdot 181^{3} + \left(145 a^{2} + 103 a + 94\right)\cdot 181^{4} + \left(60 a^{2} + 28 a + 163\right)\cdot 181^{5} + \left(59 a^{2} + 148 a + 174\right)\cdot 181^{6} + \left(46 a^{2} + 179 a + 79\right)\cdot 181^{7} + \left(15 a^{2} + 19 a + 9\right)\cdot 181^{8} + \left(66 a^{2} + 84 a + 10\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 106 a^{2} + 94 a + 27 + \left(50 a^{2} + 110 a + 63\right)\cdot 181 + \left(12 a^{2} + 9 a + 117\right)\cdot 181^{2} + \left(74 a^{2} + 134 a + 99\right)\cdot 181^{3} + \left(157 a^{2} + 23 a + 80\right)\cdot 181^{4} + \left(144 a^{2} + 89 a + 80\right)\cdot 181^{5} + \left(105 a^{2} + 123 a + 44\right)\cdot 181^{6} + \left(163 a^{2} + 99 a + 153\right)\cdot 181^{7} + \left(144 a^{2} + 58 a + 110\right)\cdot 181^{8} + \left(84 a^{2} + 176 a + 9\right)\cdot 181^{9} +O(181^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $6$ | $2$ | $(1,6)(3,8)$ | $4$ |
| $9$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-4$ |
| $12$ | $2$ | $(1,3)$ | $-2$ |
| $24$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $36$ | $2$ | $(1,3)(2,5)(4,7)$ | $-2$ |
| $16$ | $3$ | $(3,6,8)$ | $-3$ |
| $64$ | $3$ | $(3,6,8)(4,5,7)$ | $0$ |
| $12$ | $4$ | $(1,3,6,8)$ | $2$ |
| $36$ | $4$ | $(1,3,6,8)(2,4,5,7)$ | $0$ |
| $36$ | $4$ | $(1,6)(2,4,5,7)(3,8)$ | $2$ |
| $72$ | $4$ | $(1,5,6,2)(3,7,8,4)$ | $0$ |
| $72$ | $4$ | $(1,3)(2,4,5,7)$ | $0$ |
| $144$ | $4$ | $(1,4,3,2)(5,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(2,5)(3,8,6)(4,7)$ | $1$ |
| $96$ | $6$ | $(2,4)(3,6,8)$ | $1$ |
| $192$ | $6$ | $(1,2)(3,5,6,7,8,4)$ | $0$ |
| $144$ | $8$ | $(1,4,3,5,6,7,8,2)$ | $0$ |
| $96$ | $12$ | $(2,4,5,7)(3,6,8)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.