Basic invariants
| Dimension: | $8$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(139314069504\)\(\medspace = 2^{18} \cdot 3^{12} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.58773123072.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 12T213 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.58773123072.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 6x^{7} - 12x^{6} + 12x^{5} - 6x^{4} + 12x^{3} + 6x^{2} - 3x - 17 \)
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The roots of $f$ are computed in an extension of $\Q_{ 193 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 193 }$:
\( x^{3} + x + 188 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 a^{2} + 34 a + 50 + \left(92 a^{2} + 170 a + 185\right)\cdot 193 + \left(49 a^{2} + 116 a + 178\right)\cdot 193^{2} + \left(116 a^{2} + 92 a + 57\right)\cdot 193^{3} + \left(78 a^{2} + 128 a + 80\right)\cdot 193^{4} + \left(166 a^{2} + 177 a + 83\right)\cdot 193^{5} + \left(183 a^{2} + 65 a + 20\right)\cdot 193^{6} + \left(142 a^{2} + 67 a + 153\right)\cdot 193^{7} + \left(136 a^{2} + 50 a + 105\right)\cdot 193^{8} + \left(102 a^{2} + 128 a + 97\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 a^{2} + 115 a + 121 + \left(114 a^{2} + 42 a + 135\right)\cdot 193 + \left(106 a^{2} + 7 a + 152\right)\cdot 193^{2} + \left(59 a^{2} + 8 a + 148\right)\cdot 193^{3} + \left(7 a^{2} + 57 a + 32\right)\cdot 193^{4} + \left(111 a^{2} + 117 a + 175\right)\cdot 193^{5} + \left(129 a^{2} + 37 a + 112\right)\cdot 193^{6} + \left(88 a^{2} + 117 a + 52\right)\cdot 193^{7} + \left(102 a^{2} + 71 a + 147\right)\cdot 193^{8} + \left(176 a^{2} + 190 a + 146\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 a^{2} + 130 a + 49 + \left(108 a^{2} + 104 a + 94\right)\cdot 193 + \left(154 a^{2} + 93 a + 80\right)\cdot 193^{2} + \left(184 a^{2} + 103 a + 169\right)\cdot 193^{3} + \left(73 a^{2} + 62 a + 123\right)\cdot 193^{4} + \left(116 a^{2} + 51 a + 34\right)\cdot 193^{5} + \left(172 a^{2} + 140 a + 154\right)\cdot 193^{6} + \left(22 a^{2} + 168 a + 29\right)\cdot 193^{7} + \left(160 a^{2} + 185 a + 46\right)\cdot 193^{8} + \left(180 a^{2} + 56 a + 157\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 45 a^{2} + 20 a + 65 + \left(92 a^{2} + 11 a + 19\right)\cdot 193 + \left(93 a^{2} + 135 a + 104\right)\cdot 193^{2} + \left(173 a^{2} + 175 a + 97\right)\cdot 193^{3} + \left(56 a^{2} + 174 a + 112\right)\cdot 193^{4} + \left(71 a^{2} + 69 a + 4\right)\cdot 193^{5} + \left(144 a^{2} + 92 a + 71\right)\cdot 193^{6} + \left(55 a^{2} + 152 a + 180\right)\cdot 193^{7} + \left(74 a^{2} + 115 a + 181\right)\cdot 193^{8} + \left(14 a^{2} + 186 a + 174\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 103 a^{2} + 74 a + 181 + \left(97 a^{2} + 89 a + 111\right)\cdot 193 + \left(149 a^{2} + 56 a + 40\right)\cdot 193^{2} + \left(127 a^{2} + 63 a + 187\right)\cdot 193^{3} + \left(20 a^{2} + 46 a + 39\right)\cdot 193^{4} + \left(78 a^{2} + 144 a + 58\right)\cdot 193^{5} + \left(67 a^{2} + 153 a + 172\right)\cdot 193^{6} + \left(102 a^{2} + 178 a + 188\right)\cdot 193^{7} + \left(118 a^{2} + 60 a + 124\right)\cdot 193^{8} + \left(83 a^{2} + 35 a + 118\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 123 a^{2} + 79 a + 130 + \left(135 a^{2} + 104 a + 8\right)\cdot 193 + \left(83 a^{2} + 42 a + 61\right)\cdot 193^{2} + \left(149 a^{2} + 17 a + 137\right)\cdot 193^{3} + \left(25 a^{2} + 155 a + 107\right)\cdot 193^{4} + \left(122 a^{2} + 75 a + 87\right)\cdot 193^{5} + \left(38 a^{2} + 127 a + 24\right)\cdot 193^{6} + \left(33 a^{2} + 17 a + 14\right)\cdot 193^{7} + \left(15 a^{2} + 183 a + 56\right)\cdot 193^{8} + \left(37 a^{2} + 82 a + 23\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 127 a^{2} + 43 a + 184 + \left(185 a^{2} + 77 a + 145\right)\cdot 193 + \left(137 a^{2} + 157 a + 133\right)\cdot 193^{2} + \left(27 a^{2} + 106 a + 64\right)\cdot 193^{3} + \left(62 a^{2} + 148 a + 180\right)\cdot 193^{4} + \left(5 a^{2} + 71 a + 24\right)\cdot 193^{5} + \left(69 a^{2} + 153 a + 85\right)\cdot 193^{6} + \left(114 a^{2} + 64 a + 26\right)\cdot 193^{7} + \left(151 a^{2} + 84 a + 169\right)\cdot 193^{8} + \left(190 a^{2} + 142 a + 163\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 160 a^{2} + 40 a + 26 + \left(152 a^{2} + 192 a + 20\right)\cdot 193 + \left(152 a^{2} + 93 a + 107\right)\cdot 193^{2} + \left(108 a^{2} + 112 a + 174\right)\cdot 193^{3} + \left(146 a^{2} + 184 a + 123\right)\cdot 193^{4} + \left(185 a^{2} + 165 a + 65\right)\cdot 193^{5} + \left(86 a^{2} + 104 a + 185\right)\cdot 193^{6} + \left(57 a^{2} + 189 a + 158\right)\cdot 193^{7} + \left(59 a^{2} + 141 a + 149\right)\cdot 193^{8} + \left(72 a^{2} + 74 a + 46\right)\cdot 193^{9} +O(193^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 173 a^{2} + 44 a + 162 + \left(179 a^{2} + 173 a + 50\right)\cdot 193 + \left(36 a^{2} + 68 a + 106\right)\cdot 193^{2} + \left(17 a^{2} + 92 a + 120\right)\cdot 193^{3} + \left(107 a^{2} + 7 a + 163\right)\cdot 193^{4} + \left(108 a^{2} + 91 a + 44\right)\cdot 193^{5} + \left(72 a^{2} + 89 a + 139\right)\cdot 193^{6} + \left(154 a^{2} + 8 a + 160\right)\cdot 193^{7} + \left(146 a^{2} + 71 a + 176\right)\cdot 193^{8} + \left(106 a^{2} + 67 a + 35\right)\cdot 193^{9} +O(193^{10})\)
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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 |
| $1$ | $1$ | $()$ | $8$ |
| $9$ | $2$ | $(2,3)$ | $0$ |
| $18$ | $2$ | $(2,7)(3,8)(5,9)$ | $4$ |
| $27$ | $2$ | $(1,4)(2,3)(7,8)$ | $0$ |
| $27$ | $2$ | $(2,3)(7,8)$ | $0$ |
| $54$ | $2$ | $(1,7)(2,3)(4,8)(6,9)$ | $0$ |
| $6$ | $3$ | $(1,4,6)$ | $-4$ |
| $8$ | $3$ | $(1,4,6)(2,3,5)(7,8,9)$ | $-1$ |
| $12$ | $3$ | $(1,4,6)(2,3,5)$ | $2$ |
| $72$ | $3$ | $(1,2,7)(3,8,4)(5,9,6)$ | $2$ |
| $54$ | $4$ | $(2,8,3,7)(5,9)$ | $0$ |
| $162$ | $4$ | $(1,2,4,3)(5,6)(8,9)$ | $0$ |
| $36$ | $6$ | $(1,4,6)(2,7)(3,8)(5,9)$ | $-2$ |
| $36$ | $6$ | $(1,3,4,5,6,2)$ | $-2$ |
| $36$ | $6$ | $(1,4,6)(2,3)$ | $0$ |
| $36$ | $6$ | $(1,4,6)(2,3)(7,8,9)$ | $0$ |
| $54$ | $6$ | $(1,6,4)(2,3)(7,8)$ | $0$ |
| $72$ | $6$ | $(1,4,6)(2,7,3,8,5,9)$ | $1$ |
| $108$ | $6$ | $(1,8,4,9,6,7)(2,3)$ | $0$ |
| $216$ | $6$ | $(1,2,8,4,3,7)(5,9,6)$ | $0$ |
| $144$ | $9$ | $(1,3,8,4,5,9,6,2,7)$ | $-1$ |
| $108$ | $12$ | $(1,4,6)(2,8,3,7)(5,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.