Basic invariants
| Dimension: | $6$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(329845952\)\(\medspace = 2^{6} \cdot 31^{3} \cdot 173 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.10225224512.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T319 |
| Parity: | odd |
| Determinant: | 1.5363.2t1.a.a |
| Projective image: | $S_3\wr S_3$ |
| Projective stem field: | Galois closure of 9.1.10225224512.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 5x^{7} - 3x^{6} + 15x^{5} - 6x^{4} + 4x^{3} - 9x^{2} + 5x - 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$:
\( x^{3} + 2x + 225 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 178 + 46\cdot 227 + 122\cdot 227^{2} + 75\cdot 227^{3} + 137\cdot 227^{4} + 148\cdot 227^{5} + 105\cdot 227^{6} + 174\cdot 227^{7} + 85\cdot 227^{8} + 32\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 184 + 175\cdot 227 + 38\cdot 227^{2} + 185\cdot 227^{3} + 137\cdot 227^{4} + 182\cdot 227^{5} + 51\cdot 227^{6} + 171\cdot 227^{7} + 48\cdot 227^{8} + 49\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 197 + 210\cdot 227 + 213\cdot 227^{2} + 148\cdot 227^{3} + 22\cdot 227^{4} + 35\cdot 227^{5} + 155\cdot 227^{6} + 181\cdot 227^{7} + 109\cdot 227^{8} + 30\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 54 a^{2} + 166 a + 62 + \left(143 a^{2} + 201 a + 156\right)\cdot 227 + \left(223 a^{2} + 100 a + 225\right)\cdot 227^{2} + \left(49 a^{2} + a + 104\right)\cdot 227^{3} + \left(226 a^{2} + 67 a + 3\right)\cdot 227^{4} + \left(21 a^{2} + 116 a + 193\right)\cdot 227^{5} + \left(13 a^{2} + 127 a + 168\right)\cdot 227^{6} + \left(196 a^{2} + 2 a + 139\right)\cdot 227^{7} + \left(199 a^{2} + 214 a + 57\right)\cdot 227^{8} + \left(56 a^{2} + 221 a + 123\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 a^{2} + 122 a + 78 + \left(208 a^{2} + 186 a + 167\right)\cdot 227 + \left(128 a^{2} + 37 a + 23\right)\cdot 227^{2} + \left(54 a^{2} + 93 a + 111\right)\cdot 227^{3} + \left(86 a^{2} + 25 a + 119\right)\cdot 227^{4} + \left(223 a^{2} + 17 a + 7\right)\cdot 227^{5} + \left(115 a^{2} + 127 a + 79\right)\cdot 227^{6} + \left(8 a^{2} + 224 a + 192\right)\cdot 227^{7} + \left(136 a^{2} + 106 a + 123\right)\cdot 227^{8} + \left(43 a^{2} + 208 a + 105\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 87 a^{2} + 83 a + 16 + \left(181 a^{2} + 191 a + 132\right)\cdot 227 + \left(141 a^{2} + 13 a + 136\right)\cdot 227^{2} + \left(22 a^{2} + 49 a + 6\right)\cdot 227^{3} + \left(194 a^{2} + 143 a + 155\right)\cdot 227^{4} + \left(133 a^{2} + 157 a + 119\right)\cdot 227^{5} + \left(24 a^{2} + 141 a + 155\right)\cdot 227^{6} + \left(4 a^{2} + 175 a + 102\right)\cdot 227^{7} + \left(95 a^{2} + 21 a + 178\right)\cdot 227^{8} + \left(114 a^{2} + 125 a + 67\right)\cdot 227^{9} +O(227^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 107 a^{2} + 166 a + 57 + \left(102 a^{2} + 65 a + 26\right)\cdot 227 + \left(101 a^{2} + 88 a + 214\right)\cdot 227^{2} + \left(122 a^{2} + 132 a + 125\right)\cdot 227^{3} + \left(141 a^{2} + 134 a + 117\right)\cdot 227^{4} + \left(208 a^{2} + 93 a + 63\right)\cdot 227^{5} + \left(97 a^{2} + 199 a + 206\right)\cdot 227^{6} + \left(22 a^{2} + 226 a + 210\right)\cdot 227^{7} + \left(118 a^{2} + 132 a + 99\right)\cdot 227^{8} + \left(126 a^{2} + 23 a + 140\right)\cdot 227^{9} +O(227^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 142 a^{2} + 106 a + 165 + \left(115 a^{2} + 35 a + 195\right)\cdot 227 + \left(57 a^{2} + 112 a + 99\right)\cdot 227^{2} + \left(108 a^{2} + 190 a + 196\right)\cdot 227^{3} + \left(56 a^{2} + 54 a + 122\right)\cdot 227^{4} + \left(173 a^{2} + 139 a + 96\right)\cdot 227^{5} + \left(73 a^{2} + 164 a + 145\right)\cdot 227^{6} + \left(88 a^{2} + 149 a + 63\right)\cdot 227^{7} + \left(187 a^{2} + 111 a + 150\right)\cdot 227^{8} + \left(200 a^{2} + 216 a + 31\right)\cdot 227^{9} +O(227^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 225 a^{2} + 38 a + 200 + \left(156 a^{2} + 23\right)\cdot 227 + \left(27 a^{2} + 101 a + 60\right)\cdot 227^{2} + \left(96 a^{2} + 214 a + 180\right)\cdot 227^{3} + \left(203 a^{2} + 28 a + 91\right)\cdot 227^{4} + \left(146 a^{2} + 157 a + 61\right)\cdot 227^{5} + \left(128 a^{2} + 147 a + 67\right)\cdot 227^{6} + \left(134 a^{2} + 128 a + 125\right)\cdot 227^{7} + \left(171 a^{2} + 93 a + 53\right)\cdot 227^{8} + \left(138 a^{2} + 112 a + 100\right)\cdot 227^{9} +O(227^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)$ | $4$ |
| $18$ | $2$ | $(1,4)(2,5)(3,7)$ | $-2$ |
| $27$ | $2$ | $(1,2)(4,5)(6,8)$ | $0$ |
| $27$ | $2$ | $(1,2)(4,5)$ | $2$ |
| $54$ | $2$ | $(1,2)(4,6)(5,8)(7,9)$ | $0$ |
| $6$ | $3$ | $(6,8,9)$ | $3$ |
| $8$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $-3$ |
| $12$ | $3$ | $(4,5,7)(6,8,9)$ | $0$ |
| $72$ | $3$ | $(1,4,6)(2,5,8)(3,7,9)$ | $0$ |
| $54$ | $4$ | $(1,5,2,4)(3,7)$ | $-2$ |
| $162$ | $4$ | $(1,8,2,6)(3,9)(5,7)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,5)(3,7)(6,8,9)$ | $1$ |
| $36$ | $6$ | $(1,6,2,8,3,9)$ | $-2$ |
| $36$ | $6$ | $(1,2)(6,8,9)$ | $1$ |
| $36$ | $6$ | $(1,2)(4,5,7)(6,8,9)$ | $-2$ |
| $54$ | $6$ | $(1,2)(4,5)(6,9,8)$ | $-1$ |
| $72$ | $6$ | $(1,4,2,5,3,7)(6,8,9)$ | $1$ |
| $108$ | $6$ | $(1,2)(4,6,5,8,7,9)$ | $0$ |
| $216$ | $6$ | $(1,5,8,2,4,6)(3,7,9)$ | $0$ |
| $144$ | $9$ | $(1,4,6,2,5,8,3,7,9)$ | $0$ |
| $108$ | $12$ | $(1,5,2,4)(3,7)(6,8,9)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.