Basic invariants
| Dimension: | $18$ |
| Group: | $S_4\wr C_2$ |
| Conductor: | \(747\!\cdots\!000\)\(\medspace = 2^{12} \cdot 3^{9} \cdot 5^{12} \cdot 11^{14} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 8.2.32612827500.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 36T1758 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $S_4\wr C_2$ |
| Projective stem field: | Galois closure of 8.2.32612827500.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 3x^{7} - 4x^{6} + 9x^{5} + 11x^{4} + 3x^{3} - 2x^{2} + 3x - 2 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 34 + 157\cdot 181 + 66\cdot 181^{2} + 85\cdot 181^{3} + 131\cdot 181^{4} + 111\cdot 181^{5} + 90\cdot 181^{6} + 170\cdot 181^{7} + 44\cdot 181^{8} + 32\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 155 + 78\cdot 181 + 148\cdot 181^{2} + 111\cdot 181^{3} + 145\cdot 181^{4} + 24\cdot 181^{5} + 116\cdot 181^{6} + 104\cdot 181^{7} + 39\cdot 181^{8} + 128\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 15 a^{2} + 18 a + 35 + \left(91 a^{2} + 159 a + 72\right)\cdot 181 + \left(88 a^{2} + 168 a + 168\right)\cdot 181^{2} + \left(126 a^{2} + 150 a + 4\right)\cdot 181^{3} + \left(95 a^{2} + 99 a + 19\right)\cdot 181^{4} + \left(166 a^{2} + 73 a + 100\right)\cdot 181^{5} + \left(63 a^{2} + 43 a + 165\right)\cdot 181^{6} + \left(90 a^{2} + 43 a + 112\right)\cdot 181^{7} + \left(177 a^{2} + 25 a + 164\right)\cdot 181^{8} + \left(150 a^{2} + 110 a + 53\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 15 a^{2} + 178 a + 35 + \left(38 a^{2} + 132 a + 41\right)\cdot 181 + \left(21 a^{2} + 86 a + 80\right)\cdot 181^{2} + \left(157 a^{2} + 37 a + 127\right)\cdot 181^{3} + \left(82 a^{2} + 106 a + 148\right)\cdot 181^{4} + \left(a^{2} + 108 a + 163\right)\cdot 181^{5} + \left(138 a^{2} + 81 a + 99\right)\cdot 181^{6} + \left(76 a^{2} + 106 a + 58\right)\cdot 181^{7} + \left(154 a^{2} + 93 a + 72\right)\cdot 181^{8} + \left(65 a^{2} + 64 a + 75\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 82 a^{2} + 119 a + 110 + \left(160 a^{2} + 39 a + 10\right)\cdot 181 + \left(63 a^{2} + 34 a + 128\right)\cdot 181^{2} + \left(138 a^{2} + 97 a + 83\right)\cdot 181^{3} + \left(97 a^{2} + 48 a + 119\right)\cdot 181^{4} + \left(25 a^{2} + 120 a + 79\right)\cdot 181^{5} + \left(119 a^{2} + 128 a + 75\right)\cdot 181^{6} + \left(23 a^{2} + 53 a + 70\right)\cdot 181^{7} + \left(145 a^{2} + 72 a + 132\right)\cdot 181^{8} + \left(120 a^{2} + 122 a + 74\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 128 a^{2} + 108 a + 113 + \left(161 a^{2} + 143 a + 15\right)\cdot 181 + \left(10 a^{2} + 173 a + 97\right)\cdot 181^{2} + \left(98 a^{2} + 114 a + 103\right)\cdot 181^{3} + \left(180 a^{2} + 57 a + 88\right)\cdot 181^{4} + \left(a^{2} + 3 a + 166\right)\cdot 181^{5} + \left(141 a^{2} + 35 a + 162\right)\cdot 181^{6} + \left(19 a^{2} + 158 a + 54\right)\cdot 181^{7} + \left(57 a^{2} + 171 a + 142\right)\cdot 181^{8} + \left(97 a^{2} + 58 a + 161\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 151 a^{2} + 166 a + 36 + \left(51 a^{2} + 69 a + 96\right)\cdot 181 + \left(71 a^{2} + 106 a + 99\right)\cdot 181^{2} + \left(78 a^{2} + 173 a + 174\right)\cdot 181^{3} + \left(2 a^{2} + 155 a + 7\right)\cdot 181^{4} + \left(13 a^{2} + 179 a + 29\right)\cdot 181^{5} + \left(160 a^{2} + 55 a + 7\right)\cdot 181^{6} + \left(13 a^{2} + 31 a + 169\right)\cdot 181^{7} + \left(30 a^{2} + 62 a + 117\right)\cdot 181^{8} + \left(145 a^{2} + 6 a + 30\right)\cdot 181^{9} +O(181^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 152 a^{2} + 135 a + 28 + \left(39 a^{2} + 178 a + 71\right)\cdot 181 + \left(106 a^{2} + 153 a + 116\right)\cdot 181^{2} + \left(125 a^{2} + 149 a + 32\right)\cdot 181^{3} + \left(83 a^{2} + 74 a + 63\right)\cdot 181^{4} + \left(153 a^{2} + 57 a + 48\right)\cdot 181^{5} + \left(101 a^{2} + 17 a + 6\right)\cdot 181^{6} + \left(137 a^{2} + 150 a + 164\right)\cdot 181^{7} + \left(159 a^{2} + 117 a + 9\right)\cdot 181^{8} + \left(143 a^{2} + 180 a + 167\right)\cdot 181^{9} +O(181^{10})\)
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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 |
| $1$ | $1$ | $()$ | $18$ |
| $6$ | $2$ | $(2,4)(3,7)$ | $-6$ |
| $9$ | $2$ | $(1,6)(2,4)(3,7)(5,8)$ | $2$ |
| $12$ | $2$ | $(2,3)$ | $0$ |
| $24$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $36$ | $2$ | $(1,5)(2,3)$ | $-2$ |
| $36$ | $2$ | $(1,6)(2,3)(5,8)$ | $0$ |
| $16$ | $3$ | $(3,4,7)$ | $0$ |
| $64$ | $3$ | $(2,4,7)(5,6,8)$ | $0$ |
| $12$ | $4$ | $(2,3,4,7)$ | $0$ |
| $36$ | $4$ | $(1,5,6,8)(2,3,4,7)$ | $-2$ |
| $36$ | $4$ | $(1,5,6,8)(2,4)(3,7)$ | $0$ |
| $72$ | $4$ | $(1,2,6,4)(3,8,7,5)$ | $0$ |
| $72$ | $4$ | $(1,5,6,8)(2,3)$ | $2$ |
| $144$ | $4$ | $(1,2,5,3)(4,6)(7,8)$ | $0$ |
| $48$ | $6$ | $(1,6)(3,7,4)(5,8)$ | $0$ |
| $96$ | $6$ | $(2,3)(5,8,6)$ | $0$ |
| $192$ | $6$ | $(1,3)(2,5,4,6,7,8)$ | $0$ |
| $144$ | $8$ | $(1,2,5,3,6,4,8,7)$ | $0$ |
| $96$ | $12$ | $(1,5,6,8)(3,4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.