Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(191160000\)\(\medspace = 2^{6} \cdot 3^{4} \cdot 5^{4} \cdot 59 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.6372000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.59.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.6372000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 2x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 2x^{2} + 3x + 1 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 4 a^{2} + 117 a + 49 + \left(29 a^{2} + 63 a + 157\right)\cdot 181 + \left(107 a^{2} + 13 a + 53\right)\cdot 181^{2} + \left(102 a^{2} + 48 a + 133\right)\cdot 181^{3} + \left(36 a^{2} + 156 a + 48\right)\cdot 181^{4} + \left(2 a^{2} + 25 a\right)\cdot 181^{5} + \left(70 a^{2} + 17 a + 19\right)\cdot 181^{6} + \left(37 a^{2} + 28 a + 137\right)\cdot 181^{7} + \left(45 a^{2} + 54 a + 174\right)\cdot 181^{8} + \left(152 a^{2} + 95 a + 105\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 a^{2} + 37 a + 151 + \left(111 a^{2} + 44 a + 122\right)\cdot 181 + \left(32 a^{2} + 170 a + 63\right)\cdot 181^{2} + \left(90 a^{2} + 155 a + 17\right)\cdot 181^{3} + \left(172 a^{2} + 92 a + 64\right)\cdot 181^{4} + \left(64 a^{2} + 176 a + 111\right)\cdot 181^{5} + \left(149 a^{2} + 176 a + 21\right)\cdot 181^{6} + \left(127 a^{2} + 36 a + 4\right)\cdot 181^{7} + \left(20 a^{2} + 66 a + 8\right)\cdot 181^{8} + \left(154 a^{2} + 71 a + 125\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 7 a^{2} + 38 a + 174 + \left(18 a^{2} + 154 a + 110\right)\cdot 181 + \left(89 a^{2} + 13\right)\cdot 181^{2} + \left(130 a^{2} + 75 a + 177\right)\cdot 181^{3} + \left(114 a^{2} + 109 a + 35\right)\cdot 181^{4} + \left(33 a^{2} + 72 a + 50\right)\cdot 181^{5} + \left(66 a^{2} + 119 a + 136\right)\cdot 181^{6} + \left(107 a^{2} + 140 a + 164\right)\cdot 181^{7} + \left(91 a^{2} + 93 a + 24\right)\cdot 181^{8} + \left(113 a^{2} + 65 a + 121\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 26 a^{2} + 143 a + 69 + \left(72 a^{2} + 72 a + 146\right)\cdot 181 + \left(49 a^{2} + 145 a + 35\right)\cdot 181^{2} + \left(33 a^{2} + 85 a + 150\right)\cdot 181^{3} + \left(88 a + 120\right)\cdot 181^{4} + \left(82 a^{2} + 13 a + 62\right)\cdot 181^{5} + \left(140 a^{2} + 71 a + 71\right)\cdot 181^{6} + \left(88 a^{2} + 117 a + 90\right)\cdot 181^{7} + \left(13 a^{2} + 11 a + 74\right)\cdot 181^{8} + \left(155 a^{2} + 89 a + 106\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 36 a^{2} + 161 a + 177 + \left(168 a^{2} + 81 a + 170\right)\cdot 181 + \left(49 a^{2} + 15 a + 5\right)\cdot 181^{2} + \left(154 a^{2} + 6 a + 159\right)\cdot 181^{3} + \left(138 a^{2} + 163 a + 95\right)\cdot 181^{4} + \left(118 a^{2} + 99 a + 104\right)\cdot 181^{5} + \left(174 a^{2} + 57 a + 75\right)\cdot 181^{6} + \left(50 a^{2} + 2 a + 10\right)\cdot 181^{7} + \left(76 a^{2} + 69 a + 118\right)\cdot 181^{8} + \left(29 a^{2} + 180 a + 157\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 54 a^{2} + 67 a + 158 + \left(56 a^{2} + 114 a + 84\right)\cdot 181 + \left(147 a^{2} + 29 a + 160\right)\cdot 181^{2} + \left(89 a^{2} + 5 a + 15\right)\cdot 181^{3} + \left(131 a^{2} + 105 a + 81\right)\cdot 181^{4} + \left(8 a^{2} + 32 a + 67\right)\cdot 181^{5} + \left(163 a^{2} + 119 a + 76\right)\cdot 181^{6} + \left(9 a^{2} + 149 a + 75\right)\cdot 181^{7} + \left(18 a^{2} + 16 a + 178\right)\cdot 181^{8} + \left(81 a^{2} + 120 a + 13\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 120 a^{2} + 77 a + 60 + \left(13 a^{2} + 22 a + 95\right)\cdot 181 + \left(a^{2} + 162 a + 118\right)\cdot 181^{2} + \left(a^{2} + 19 a + 22\right)\cdot 181^{3} + \left(58 a^{2} + 164 a + 149\right)\cdot 181^{4} + \left(107 a^{2} + 152 a + 99\right)\cdot 181^{5} + \left(49 a^{2} + 65 a + 165\right)\cdot 181^{6} + \left(43 a^{2} + 175 a + 27\right)\cdot 181^{7} + \left(142 a^{2} + 97 a + 132\right)\cdot 181^{8} + \left(126 a^{2} + 170 a + 15\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 141 a^{2} + 84 a + 54 + \left(164 a^{2} + 35 a + 157\right)\cdot 181 + \left(23 a^{2} + 152 a + 82\right)\cdot 181^{2} + \left(105 a^{2} + 126 a + 143\right)\cdot 181^{3} + \left(5 a^{2} + 42 a + 105\right)\cdot 181^{4} + \left(60 a^{2} + 55 a + 50\right)\cdot 181^{5} + \left(117 a^{2} + 106 a + 27\right)\cdot 181^{6} + \left(92 a^{2} + 150 a + 177\right)\cdot 181^{7} + \left(59 a^{2} + 57 a + 50\right)\cdot 181^{8} + \left(180 a^{2} + 86 a + 37\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 148 a^{2} + 14 + \left(90 a^{2} + 135 a + 40\right)\cdot 181 + \left(42 a^{2} + 34 a + 8\right)\cdot 181^{2} + \left(17 a^{2} + 20 a + 86\right)\cdot 181^{3} + \left(66 a^{2} + 164 a + 22\right)\cdot 181^{4} + \left(65 a^{2} + 94 a + 177\right)\cdot 181^{5} + \left(155 a^{2} + 171 a + 130\right)\cdot 181^{6} + \left(165 a^{2} + 103 a + 36\right)\cdot 181^{7} + \left(75 a^{2} + 75 a + 143\right)\cdot 181^{8} + \left(93 a^{2} + 26 a + 40\right)\cdot 181^{9} +O(181^{10})\)
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Generators of the action on the roots $r_1, \ldots, r_{ 9 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 9 }$ | Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,4)$ | $4$ |
| $18$ | $2$ | $(1,3)(4,6)(7,8)$ | $-2$ |
| $27$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $27$ | $2$ | $(1,4)(3,6)$ | $2$ |
| $54$ | $2$ | $(1,4)(2,3)(5,6)(8,9)$ | $0$ |
| $6$ | $3$ | $(2,5,9)$ | $3$ |
| $8$ | $3$ | $(1,4,7)(2,5,9)(3,6,8)$ | $-3$ |
| $12$ | $3$ | $(2,5,9)(3,6,8)$ | $0$ |
| $72$ | $3$ | $(1,3,2)(4,6,5)(7,8,9)$ | $0$ |
| $54$ | $4$ | $(1,6,4,3)(7,8)$ | $-2$ |
| $162$ | $4$ | $(1,5,4,2)(6,8)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,3)(2,5,9)(4,6)(7,8)$ | $1$ |
| $36$ | $6$ | $(1,2,4,5,7,9)$ | $-2$ |
| $36$ | $6$ | $(1,4)(2,5,9)$ | $1$ |
| $36$ | $6$ | $(1,4)(2,5,9)(3,6,8)$ | $-2$ |
| $54$ | $6$ | $(1,4)(2,9,5)(3,6)$ | $-1$ |
| $72$ | $6$ | $(1,3,4,6,7,8)(2,5,9)$ | $1$ |
| $108$ | $6$ | $(1,4)(2,6,5,8,9,3)$ | $0$ |
| $216$ | $6$ | $(1,6,5,4,3,2)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,2,4,6,5,7,8,9)$ | $0$ |
| $108$ | $12$ | $(1,6,4,3)(2,5,9)(7,8)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.