Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(49443696\)\(\medspace = 2^{4} \cdot 3^{9} \cdot 157 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.5339919168.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.471.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.5339919168.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{7} - 3x^{6} + 3x^{5} + 6x^{4} + 6x^{3} - 3x^{2} - 3x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{3} + x + 196 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 a^{2} + 137 a + 2 + \left(84 a^{2} + 147 a + 147\right)\cdot 199 + \left(194 a^{2} + 157 a + 1\right)\cdot 199^{2} + \left(23 a^{2} + 128 a + 128\right)\cdot 199^{3} + \left(14 a^{2} + 138 a + 3\right)\cdot 199^{4} + \left(76 a^{2} + 9 a + 90\right)\cdot 199^{5} + \left(a^{2} + 82 a + 117\right)\cdot 199^{6} + \left(161 a^{2} + 52 a + 90\right)\cdot 199^{7} + \left(79 a^{2} + 132 a + 115\right)\cdot 199^{8} + \left(66 a^{2} + 8 a + 29\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 63 a^{2} + 27 a + 166 + \left(47 a^{2} + 13 a + 188\right)\cdot 199 + \left(178 a + 4\right)\cdot 199^{2} + \left(193 a^{2} + 31 a + 108\right)\cdot 199^{3} + \left(163 a^{2} + 48 a + 103\right)\cdot 199^{4} + \left(175 a^{2} + 167 a + 156\right)\cdot 199^{5} + \left(23 a^{2} + 46 a + 198\right)\cdot 199^{6} + \left(105 a^{2} + 2 a + 185\right)\cdot 199^{7} + \left(125 a^{2} + 136 a + 145\right)\cdot 199^{8} + \left(76 a^{2} + 2 a + 102\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 69 a^{2} + 65 a + 73 + \left(38 a^{2} + 110 a + 75\right)\cdot 199 + \left(174 a^{2} + 27 a + 147\right)\cdot 199^{2} + \left(30 a^{2} + 150 a + 190\right)\cdot 199^{3} + \left(115 a^{2} + 88 a + 105\right)\cdot 199^{4} + \left(143 a^{2} + 7 a + 53\right)\cdot 199^{5} + \left(48 a^{2} + 110 a + 147\right)\cdot 199^{6} + \left(138 a^{2} + 165 a + 78\right)\cdot 199^{7} + \left(62 a^{2} + 104 a + 47\right)\cdot 199^{8} + \left(97 a^{2} + 46 a + 191\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 114 a^{2} + 196 a + 124 + \left(76 a^{2} + 139 a + 175\right)\cdot 199 + \left(29 a^{2} + 13 a + 49\right)\cdot 199^{2} + \left(144 a^{2} + 119 a + 79\right)\cdot 199^{3} + \left(69 a^{2} + 170 a + 89\right)\cdot 199^{4} + \left(178 a^{2} + 181 a + 55\right)\cdot 199^{5} + \left(148 a^{2} + 6 a + 133\right)\cdot 199^{6} + \left(98 a^{2} + 180 a + 29\right)\cdot 199^{7} + \left(56 a^{2} + 160 a + 36\right)\cdot 199^{8} + \left(35 a^{2} + 143 a + 177\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 120 a^{2} + 35 a + 5 + \left(67 a^{2} + 38 a + 136\right)\cdot 199 + \left(4 a^{2} + 62 a + 7\right)\cdot 199^{2} + \left(181 a^{2} + 38 a + 100\right)\cdot 199^{3} + \left(20 a^{2} + 12 a + 74\right)\cdot 199^{4} + \left(146 a^{2} + 22 a + 70\right)\cdot 199^{5} + \left(173 a^{2} + 70 a + 33\right)\cdot 199^{6} + \left(131 a^{2} + 144 a + 71\right)\cdot 199^{7} + \left(192 a^{2} + 129 a + 124\right)\cdot 199^{8} + \left(55 a^{2} + 187 a + 22\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 127 a^{2} + 137 a + \left(133 a^{2} + 158 a + 81\right)\cdot 199 + \left(120 a^{2} + 60 a + 44\right)\cdot 199^{2} + \left(138 a^{2} + 84 a + 9\right)\cdot 199^{3} + \left(189 a^{2} + 189 a + 103\right)\cdot 199^{4} + \left(107 a^{2} + 185 a + 8\right)\cdot 199^{5} + \left(103 a^{2} + 144 a + 103\right)\cdot 199^{6} + \left(160 a^{2} + 103 a + 4\right)\cdot 199^{7} + \left(155 a^{2} + 186 a + 36\right)\cdot 199^{8} + \left(38 a^{2} + 26 a + 113\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 151 a^{2} + 27 a + 194 + \left(196 a^{2} + 2 a + 180\right)\cdot 199 + \left(73 a^{2} + 76 a + 146\right)\cdot 199^{2} + \left(78 a^{2} + 76 a + 89\right)\cdot 199^{3} + \left(187 a^{2} + 196 a + 21\right)\cdot 199^{4} + \left(143 a^{2} + 189 a + 120\right)\cdot 199^{5} + \left(120 a^{2} + 182 a + 62\right)\cdot 199^{6} + \left(105 a^{2} + 149 a + 123\right)\cdot 199^{7} + \left(49 a^{2} + 81 a + 38\right)\cdot 199^{8} + \left(104 a^{2} + 183 a + 63\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 157 a^{2} + 65 a + 20 + \left(187 a^{2} + 99 a + 117\right)\cdot 199 + \left(48 a^{2} + 124 a + 195\right)\cdot 199^{2} + \left(115 a^{2} + 194 a + 59\right)\cdot 199^{3} + \left(138 a^{2} + 37 a + 135\right)\cdot 199^{4} + \left(111 a^{2} + 30 a + 143\right)\cdot 199^{5} + \left(145 a^{2} + 47 a + 64\right)\cdot 199^{6} + \left(138 a^{2} + 114 a + 56\right)\cdot 199^{7} + \left(185 a^{2} + 50 a + 122\right)\cdot 199^{8} + \left(124 a^{2} + 28 a + 170\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 178 a^{2} + 107 a + 13 + \left(162 a^{2} + 86 a + 92\right)\cdot 199 + \left(149 a^{2} + 95 a + 197\right)\cdot 199^{2} + \left(89 a^{2} + 171 a + 30\right)\cdot 199^{3} + \left(95 a^{2} + 112 a + 159\right)\cdot 199^{4} + \left(110 a^{2} + a + 97\right)\cdot 199^{5} + \left(29 a^{2} + 105 a + 134\right)\cdot 199^{6} + \left(154 a^{2} + 82 a + 155\right)\cdot 199^{7} + \left(86 a^{2} + 12 a + 129\right)\cdot 199^{8} + \left(196 a^{2} + 168 a + 124\right)\cdot 199^{9} +O(199^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(5,6)$ | $4$ |
| $18$ | $2$ | $(2,5)(6,8)(7,9)$ | $-2$ |
| $27$ | $2$ | $(1,3)(2,8)(5,6)$ | $0$ |
| $27$ | $2$ | $(2,8)(5,6)$ | $2$ |
| $54$ | $2$ | $(1,2)(3,8)(4,9)(5,6)$ | $0$ |
| $6$ | $3$ | $(1,3,4)$ | $3$ |
| $8$ | $3$ | $(1,3,4)(2,8,9)(5,6,7)$ | $-3$ |
| $12$ | $3$ | $(1,3,4)(2,8,9)$ | $0$ |
| $72$ | $3$ | $(1,5,2)(3,6,8)(4,7,9)$ | $0$ |
| $54$ | $4$ | $(2,5,8,6)(7,9)$ | $-2$ |
| $162$ | $4$ | $(1,5,3,6)(4,7)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,3,4)(2,5)(6,8)(7,9)$ | $1$ |
| $36$ | $6$ | $(1,6,3,7,4,5)$ | $-2$ |
| $36$ | $6$ | $(1,3,4)(5,6)$ | $1$ |
| $36$ | $6$ | $(1,3,4)(2,8,9)(5,6)$ | $-2$ |
| $54$ | $6$ | $(1,4,3)(2,8)(5,6)$ | $-1$ |
| $72$ | $6$ | $(1,3,4)(2,6,8,7,9,5)$ | $1$ |
| $108$ | $6$ | $(1,8,3,9,4,2)(5,6)$ | $0$ |
| $216$ | $6$ | $(1,5,8,3,6,2)(4,7,9)$ | $0$ |
| $144$ | $9$ | $(1,6,8,3,7,9,4,5,2)$ | $0$ |
| $108$ | $12$ | $(1,3,4)(2,5,8,6)(7,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.