Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(478059664\)\(\medspace = 2^{4} \cdot 29 \cdot 101^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.5.193136104256.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | even |
| Determinant: | 1.2929.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.5.193136104256.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - x^{8} - 8x^{7} - 2x^{5} - 2x^{4} + 4x^{3} - 4x^{2} + 5x - 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 197 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 197 }$:
\( x^{3} + 3x + 195 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 a^{2} + 118 a + 189 + \left(193 a^{2} + 123 a + 27\right)\cdot 197 + \left(14 a^{2} + 26\right)\cdot 197^{2} + \left(159 a^{2} + 78 a + 113\right)\cdot 197^{3} + \left(95 a^{2} + 120 a + 145\right)\cdot 197^{4} + \left(23 a^{2} + 175 a + 67\right)\cdot 197^{5} + \left(69 a^{2} + 89 a + 144\right)\cdot 197^{6} + \left(160 a^{2} + 157 a + 38\right)\cdot 197^{7} + \left(134 a^{2} + 88 a + 153\right)\cdot 197^{8} + \left(20 a^{2} + 54 a + 187\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 39 a^{2} + 158 a + 24 + \left(148 a^{2} + 95 a + 135\right)\cdot 197 + \left(23 a^{2} + 83 a + 43\right)\cdot 197^{2} + \left(141 a^{2} + 123 a + 77\right)\cdot 197^{3} + \left(23 a^{2} + 81 a + 1\right)\cdot 197^{4} + \left(155 a^{2} + 171 a + 134\right)\cdot 197^{5} + \left(166 a^{2} + 187 a + 142\right)\cdot 197^{6} + \left(174 a^{2} + 135 a + 67\right)\cdot 197^{7} + \left(139 a^{2} + 59 a + 163\right)\cdot 197^{8} + \left(195 a^{2} + 181 a + 143\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 67 a^{2} + 173 a + 120 + \left(106 a^{2} + 64 a + 164\right)\cdot 197 + \left(99 a^{2} + 110 a + 77\right)\cdot 197^{2} + \left(142 a^{2} + 149 a + 167\right)\cdot 197^{3} + \left(5 a^{2} + 77 a + 89\right)\cdot 197^{4} + \left(137 a^{2} + 29 a + 29\right)\cdot 197^{5} + \left(161 a^{2} + 72 a + 117\right)\cdot 197^{6} + \left(51 a^{2} + 100 a + 142\right)\cdot 197^{7} + \left(131 a^{2} + 76 a + 22\right)\cdot 197^{8} + \left(49 a^{2} + 45 a + 145\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 113 a^{2} + 168 a + 163 + \left(172 a^{2} + 111 a + 94\right)\cdot 197 + \left(61 a^{2} + 159 a + 117\right)\cdot 197^{2} + \left(8 a^{2} + 175 a + 76\right)\cdot 197^{3} + \left(76 a^{2} + 58 a + 185\right)\cdot 197^{4} + \left(164 a^{2} + 6 a + 92\right)\cdot 197^{5} + \left(153 a^{2} + 173 a + 179\right)\cdot 197^{6} + \left(40 a^{2} + 89 a + 61\right)\cdot 197^{7} + \left(128 a^{2} + 24 a + 87\right)\cdot 197^{8} + \left(104 a^{2} + 67 a + 148\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 131 a^{2} + 154 a + 51 + \left(148 a^{2} + 95 a + 52\right)\cdot 197 + \left(83 a^{2} + 65 a + 46\right)\cdot 197^{2} + \left(86 a^{2} + 125 a + 55\right)\cdot 197^{3} + \left(121 a^{2} + 3 a + 124\right)\cdot 197^{4} + \left(136 a^{2} + 76 a + 28\right)\cdot 197^{5} + \left(17 a^{2} + 51 a + 26\right)\cdot 197^{6} + \left(162 a^{2} + 168 a + 166\right)\cdot 197^{7} + \left(43 a^{2} + 17 a + 44\right)\cdot 197^{8} + \left(124 a^{2} + 179 a + 97\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 132 a^{2} + 143 a + 4 + \left(64 a^{2} + 115 a + 76\right)\cdot 197 + \left(112 a^{2} + 187 a + 21\right)\cdot 197^{2} + \left(156 a^{2} + 16 a + 176\right)\cdot 197^{3} + \left(180 a^{2} + 150 a\right)\cdot 197^{4} + \left(100 a^{2} + 137 a + 163\right)\cdot 197^{5} + \left(33 a^{2} + 74 a + 135\right)\cdot 197^{6} + \left(66 a^{2} + 53 a + 112\right)\cdot 197^{7} + \left(12 a^{2} + 41 a + 52\right)\cdot 197^{8} + \left(165 a^{2} + 36 a + 72\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 135 a^{2} + 118 a + 19 + \left(52 a^{2} + 174 a + 141\right)\cdot 197 + \left(158 a^{2} + 112 a + 115\right)\cdot 197^{2} + \left(93 a^{2} + 192 a + 179\right)\cdot 197^{3} + \left(77 a^{2} + 191 a + 108\right)\cdot 197^{4} + \left(18 a^{2} + 46 a + 57\right)\cdot 197^{5} + \left(158 a^{2} + 116 a + 125\right)\cdot 197^{6} + \left(58 a^{2} + 100 a + 32\right)\cdot 197^{7} + \left(119 a^{2} + 48 a + 122\right)\cdot 197^{8} + \left(177 a^{2} + 158 a + 107\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 149 a^{2} + 83 a + 38 + \left(156 a^{2} + 166 a + 63\right)\cdot 197 + \left(22 a^{2} + 46 a + 39\right)\cdot 197^{2} + \left(32 a^{2} + 4 a + 124\right)\cdot 197^{3} + \left(137 a^{2} + 185 a + 110\right)\cdot 197^{4} + \left(128 a^{2} + 52 a + 21\right)\cdot 197^{5} + \left(9 a^{2} + 146 a + 88\right)\cdot 197^{6} + \left(90 a^{2} + 53 a + 160\right)\cdot 197^{7} + \left(56 a^{2} + 131 a + 140\right)\cdot 197^{8} + \left(124 a^{2} + 93 a + 187\right)\cdot 197^{9} +O(197^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 196 a^{2} + 67 a + 181 + \left(138 a^{2} + 36 a + 32\right)\cdot 197 + \left(13 a^{2} + 21 a + 103\right)\cdot 197^{2} + \left(165 a^{2} + 119 a + 15\right)\cdot 197^{3} + \left(69 a^{2} + 115 a + 21\right)\cdot 197^{4} + \left(120 a^{2} + 91 a + 193\right)\cdot 197^{5} + \left(17 a^{2} + 73 a + 25\right)\cdot 197^{6} + \left(180 a^{2} + 125 a + 5\right)\cdot 197^{7} + \left(21 a^{2} + 102 a + 1\right)\cdot 197^{8} + \left(23 a^{2} + 169 a + 92\right)\cdot 197^{9} +O(197^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,5)$ | $4$ |
| $18$ | $2$ | $(1,2)(4,5)(8,9)$ | $-2$ |
| $27$ | $2$ | $(1,4)(2,5)(3,6)$ | $0$ |
| $27$ | $2$ | $(1,4)(2,5)$ | $2$ |
| $54$ | $2$ | $(1,3)(2,5)(4,6)(7,9)$ | $0$ |
| $6$ | $3$ | $(3,6,7)$ | $3$ |
| $8$ | $3$ | $(1,4,9)(2,5,8)(3,6,7)$ | $-3$ |
| $12$ | $3$ | $(1,4,9)(3,6,7)$ | $0$ |
| $72$ | $3$ | $(1,3,2)(4,6,5)(7,8,9)$ | $0$ |
| $54$ | $4$ | $(1,2,4,5)(8,9)$ | $-2$ |
| $162$ | $4$ | $(2,6,5,3)(4,9)(7,8)$ | $0$ |
| $36$ | $6$ | $(1,2)(3,6,7)(4,5)(8,9)$ | $1$ |
| $36$ | $6$ | $(2,3,5,6,8,7)$ | $-2$ |
| $36$ | $6$ | $(2,5)(3,6,7)$ | $1$ |
| $36$ | $6$ | $(1,4,9)(2,5)(3,6,7)$ | $-2$ |
| $54$ | $6$ | $(1,4)(2,5)(3,7,6)$ | $-1$ |
| $72$ | $6$ | $(1,5,4,8,9,2)(3,6,7)$ | $1$ |
| $108$ | $6$ | $(1,3,4,6,9,7)(2,5)$ | $0$ |
| $216$ | $6$ | $(1,3,2,4,6,5)(7,8,9)$ | $0$ |
| $144$ | $9$ | $(1,3,5,4,6,8,9,7,2)$ | $0$ |
| $108$ | $12$ | $(1,2,4,5)(3,6,7)(8,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.