Basic invariants
| Dimension: | $16$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(212\!\cdots\!304\)\(\medspace = 2^{34} \cdot 7^{8} \cdot 11^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.5877014528.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 24T2912 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.5877014528.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} + 3x^{7} - 9x^{6} + 10x^{5} - 6x^{4} + 8x^{3} - 4x^{2} - 2x + 2 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 38 + 17\cdot 199 + 152\cdot 199^{2} + 103\cdot 199^{3} + 59\cdot 199^{4} + 61\cdot 199^{5} + 66\cdot 199^{6} + 72\cdot 199^{7} + 9\cdot 199^{8} + 110\cdot 199^{9} +O(199^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 108 + 146\cdot 199 + 148\cdot 199^{2} + 17\cdot 199^{3} + 125\cdot 199^{4} + 22\cdot 199^{5} + 194\cdot 199^{6} + 53\cdot 199^{7} + 198\cdot 199^{8} + 57\cdot 199^{9} +O(199^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 166 + 144\cdot 199 + 92\cdot 199^{2} + 143\cdot 199^{3} + 92\cdot 199^{4} + 108\cdot 199^{5} + 191\cdot 199^{6} + 188\cdot 199^{7} + 191\cdot 199^{8} + 38\cdot 199^{9} +O(199^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 77 a^{2} + 138 a + 4 + \left(105 a^{2} + 46 a + 171\right)\cdot 199 + \left(18 a^{2} + 186 a + 5\right)\cdot 199^{2} + \left(180 a^{2} + 194 a + 104\right)\cdot 199^{3} + \left(32 a^{2} + 36 a + 118\right)\cdot 199^{4} + \left(5 a^{2} + 125 a + 47\right)\cdot 199^{5} + \left(14 a^{2} + 14 a + 77\right)\cdot 199^{6} + \left(123 a^{2} + 27 a + 181\right)\cdot 199^{7} + \left(77 a^{2} + 139 a + 191\right)\cdot 199^{8} + \left(12 a^{2} + 82 a + 152\right)\cdot 199^{9} +O(199^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 97 a^{2} + 100 a + 141 + \left(139 a^{2} + 52 a + 154\right)\cdot 199 + \left(49 a^{2} + 137 a + 173\right)\cdot 199^{2} + \left(97 a^{2} + 65 a + 58\right)\cdot 199^{3} + \left(51 a^{2} + 24 a + 44\right)\cdot 199^{4} + \left(111 a^{2} + 110 a + 32\right)\cdot 199^{5} + \left(65 a^{2} + 144 a + 24\right)\cdot 199^{6} + \left(94 a^{2} + 87 a + 190\right)\cdot 199^{7} + \left(151 a^{2} + 45 a + 92\right)\cdot 199^{8} + \left(138 a^{2} + 121 a + 11\right)\cdot 199^{9} +O(199^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 117 a^{2} + 13 a + 88 + \left(127 a^{2} + 171 a + 80\right)\cdot 199 + \left(52 a^{2} + a + 109\right)\cdot 199^{2} + \left(192 a^{2} + 174 a + 188\right)\cdot 199^{3} + \left(134 a^{2} + 47 a + 99\right)\cdot 199^{4} + \left(85 a^{2} + 104 a + 81\right)\cdot 199^{5} + \left(13 a^{2} + 157 a + 188\right)\cdot 199^{6} + \left(75 a^{2} + 195 a + 110\right)\cdot 199^{7} + \left(6 a^{2} + 172 a + 62\right)\cdot 199^{8} + \left(175 a^{2} + 37 a + 168\right)\cdot 199^{9} +O(199^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 144 a^{2} + 122 a + 115 + \left(4 a^{2} + 43 a + 37\right)\cdot 199 + \left(48 a^{2} + 152 a + 158\right)\cdot 199^{2} + \left(28 a^{2} + 128 a + 2\right)\cdot 199^{3} + \left(50 a^{2} + 100 a + 130\right)\cdot 199^{4} + \left(53 a^{2} + 172 a + 79\right)\cdot 199^{5} + \left(101 a^{2} + 47 a + 135\right)\cdot 199^{6} + \left(62 a^{2} + 105 a + 74\right)\cdot 199^{7} + \left(30 a^{2} + 32 a + 160\right)\cdot 199^{8} + \left(77 a^{2} + 129 a + 129\right)\cdot 199^{9} +O(199^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 177 a^{2} + 138 a + 137 + \left(88 a^{2} + 108 a + 93\right)\cdot 199 + \left(132 a^{2} + 59 a + 15\right)\cdot 199^{2} + \left(189 a^{2} + 74 a + 44\right)\cdot 199^{3} + \left(115 a^{2} + 61 a + 41\right)\cdot 199^{4} + \left(140 a^{2} + 100 a + 5\right)\cdot 199^{5} + \left(83 a^{2} + 136 a + 190\right)\cdot 199^{6} + \left(13 a^{2} + 66 a + 41\right)\cdot 199^{7} + \left(91 a^{2} + 27 a + 68\right)\cdot 199^{8} + \left(109 a^{2} + 186 a + 151\right)\cdot 199^{9} +O(199^{10})\)
| $r_{ 9 }$ |
$=$ |
\( 184 a^{2} + 86 a + \left(130 a^{2} + 174 a + 149\right)\cdot 199 + \left(96 a^{2} + 59 a + 138\right)\cdot 199^{2} + \left(108 a^{2} + 158 a + 132\right)\cdot 199^{3} + \left(12 a^{2} + 126 a + 84\right)\cdot 199^{4} + \left(2 a^{2} + 183 a + 158\right)\cdot 199^{5} + \left(120 a^{2} + 95 a + 126\right)\cdot 199^{6} + \left(29 a^{2} + 114 a + 80\right)\cdot 199^{7} + \left(41 a^{2} + 179 a + 19\right)\cdot 199^{8} + \left(84 a^{2} + 39 a + 174\right)\cdot 199^{9} +O(199^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $16$ |
| $9$ | $2$ | $(1,2)$ | $0$ |
| $18$ | $2$ | $(1,4)(2,7)(3,8)$ | $0$ |
| $27$ | $2$ | $(1,2)(4,7)(5,6)$ | $0$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,5)(2,6)(3,9)(4,7)$ | $0$ |
| $6$ | $3$ | $(5,6,9)$ | $-8$ |
| $8$ | $3$ | $(1,3,2)(4,8,7)(5,9,6)$ | $-2$ |
| $12$ | $3$ | $(1,3,2)(5,9,6)$ | $4$ |
| $72$ | $3$ | $(1,4,5)(2,7,6)(3,8,9)$ | $-2$ |
| $54$ | $4$ | $(1,5,2,6)(3,9)$ | $0$ |
| $162$ | $4$ | $(1,5,2,6)(3,9)(4,7)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,7)(3,8)(5,6,9)$ | $0$ |
| $36$ | $6$ | $(1,5,3,9,2,6)$ | $0$ |
| $36$ | $6$ | $(1,2)(5,6,9)$ | $0$ |
| $36$ | $6$ | $(1,2)(4,7,8)(5,6,9)$ | $0$ |
| $54$ | $6$ | $(1,2)(4,7)(5,9,6)$ | $0$ |
| $72$ | $6$ | $(1,4,3,8,2,7)(5,6,9)$ | $0$ |
| $108$ | $6$ | $(1,5,3,9,2,6)(4,7)$ | $0$ |
| $216$ | $6$ | $(1,4,5,2,7,6)(3,8,9)$ | $0$ |
| $144$ | $9$ | $(1,4,5,3,8,9,2,7,6)$ | $1$ |
| $108$ | $12$ | $(1,4,2,7)(3,8)(5,6,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.