Basic invariants
| Dimension: | $12$ |
| Group: | $S_3\wr S_3$ |
| Conductor: | \(332552906075897856\)\(\medspace = 2^{15} \cdot 3^{15} \cdot 29^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin number field: | Galois closure of 9.1.63126687744.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T315 |
| Parity: | odd |
| Projective image: | $S_3\wr S_3$ |
| Projective field: | Galois closure of 9.1.63126687744.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 227 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 227 }$:
\( x^{3} + 2x + 225 \)
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Roots:
| $r_{ 1 }$ | $=$ |
\( 112 + 164\cdot 227 + 164\cdot 227^{2} + 60\cdot 227^{3} + 83\cdot 227^{4} + 79\cdot 227^{5} + 46\cdot 227^{6} + 56\cdot 227^{7} + 171\cdot 227^{8} + 57\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 120 + 88\cdot 227 + 191\cdot 227^{2} + 180\cdot 227^{3} + 86\cdot 227^{4} + 12\cdot 227^{5} + 163\cdot 227^{6} + 153\cdot 227^{7} + 120\cdot 227^{8} + 182\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 163 + 202\cdot 227 + 14\cdot 227^{2} + 221\cdot 227^{3} + 37\cdot 227^{4} + 147\cdot 227^{5} + 46\cdot 227^{6} + 57\cdot 227^{7} + 23\cdot 227^{8} + 6\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 a^{2} + 77 a + 138 + \left(184 a^{2} + 180 a + 132\right)\cdot 227 + \left(76 a^{2} + 212 a + 28\right)\cdot 227^{2} + \left(85 a^{2} + 8 a + 23\right)\cdot 227^{3} + \left(124 a^{2} + 32 a + 216\right)\cdot 227^{4} + \left(38 a^{2} + 113 a + 87\right)\cdot 227^{5} + \left(14 a^{2} + 90 a + 111\right)\cdot 227^{6} + \left(225 a^{2} + 111 a\right)\cdot 227^{7} + \left(200 a^{2} + 145 a + 74\right)\cdot 227^{8} + \left(107 a^{2} + 81 a + 211\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 a^{2} + 185 a + 194 + \left(171 a^{2} + 60 a + 39\right)\cdot 227 + \left(77 a^{2} + 107 a + 181\right)\cdot 227^{2} + \left(193 a^{2} + 189 a + 15\right)\cdot 227^{3} + \left(175 a^{2} + 221 a + 209\right)\cdot 227^{4} + \left(178 a^{2} + 13 a + 47\right)\cdot 227^{5} + \left(129 a^{2} + 133 a + 114\right)\cdot 227^{6} + \left(118 a^{2} + 213 a + 85\right)\cdot 227^{7} + \left(150 a^{2} + 119 a + 82\right)\cdot 227^{8} + \left(58 a^{2} + 14 a + 221\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 65 a^{2} + 216 a + 141 + \left(45 a^{2} + 107 a + 172\right)\cdot 227 + \left(206 a^{2} + 171 a + 73\right)\cdot 227^{2} + \left(15 a^{2} + 204 a + 33\right)\cdot 227^{3} + \left(190 a^{2} + 155 a + 58\right)\cdot 227^{4} + \left(207 a^{2} + 26 a + 85\right)\cdot 227^{5} + \left(140 a^{2} + 174 a + 161\right)\cdot 227^{6} + \left(72 a^{2} + 6 a + 4\right)\cdot 227^{7} + \left(10 a^{2} + 123 a + 27\right)\cdot 227^{8} + \left(168 a^{2} + 61 a + 150\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 153 a^{2} + 192 a + 18 + \left(98 a^{2} + 212 a + 170\right)\cdot 227 + \left(72 a^{2} + 133 a + 22\right)\cdot 227^{2} + \left(175 a^{2} + 28 a + 143\right)\cdot 227^{3} + \left(153 a^{2} + 200 a + 179\right)\cdot 227^{4} + \left(9 a^{2} + 99 a + 200\right)\cdot 227^{5} + \left(83 a^{2} + 3 a + 51\right)\cdot 227^{6} + \left(110 a^{2} + 129 a + 150\right)\cdot 227^{7} + \left(102 a^{2} + 188 a + 169\right)\cdot 227^{8} + \left(60 a^{2} + 130 a + 223\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 169 a^{2} + 107 a + 204 + \left(10 a + 188\right)\cdot 227 + \left(157 a^{2} + 186 a + 83\right)\cdot 227^{2} + \left(45 a^{2} + 114 a + 224\right)\cdot 227^{3} + \left(52 a^{2} + 98 a + 176\right)\cdot 227^{4} + \left(40 a^{2} + 5 a + 88\right)\cdot 227^{5} + \left(158 a^{2} + 116 a + 184\right)\cdot 227^{6} + \left(98 a^{2} + 120 a + 190\right)\cdot 227^{7} + \left(125 a^{2} + 11 a + 104\right)\cdot 227^{8} + \left(172 a^{2} + 55 a + 80\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 220 a^{2} + 131 a + 45 + \left(180 a^{2} + 108 a + 202\right)\cdot 227 + \left(90 a^{2} + 96 a + 146\right)\cdot 227^{2} + \left(165 a^{2} + 134 a + 5\right)\cdot 227^{3} + \left(211 a^{2} + 199 a + 87\right)\cdot 227^{4} + \left(205 a^{2} + 194 a + 158\right)\cdot 227^{5} + \left(154 a^{2} + 163 a + 28\right)\cdot 227^{6} + \left(55 a^{2} + 99 a + 209\right)\cdot 227^{7} + \left(91 a^{2} + 92 a + 134\right)\cdot 227^{8} + \left(113 a^{2} + 110 a + 1\right)\cdot 227^{9} +O(227^{10})\)
|
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 values |
| $c1$ | |||
| $1$ | $1$ | $()$ | $12$ |
| $9$ | $2$ | $(4,5)$ | $4$ |
| $18$ | $2$ | $(1,4)(2,5)(3,7)$ | $2$ |
| $27$ | $2$ | $(1,2)(4,5)(6,8)$ | $0$ |
| $27$ | $2$ | $(1,2)(4,5)$ | $0$ |
| $54$ | $2$ | $(1,6)(2,8)(3,9)(4,5)$ | $2$ |
| $6$ | $3$ | $(6,8,9)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $3$ |
| $12$ | $3$ | $(1,2,3)(6,8,9)$ | $-3$ |
| $72$ | $3$ | $(1,6,4)(2,8,5)(3,9,7)$ | $0$ |
| $54$ | $4$ | $(1,4,2,5)(3,7)$ | $0$ |
| $162$ | $4$ | $(2,3)(4,8,5,6)(7,9)$ | $0$ |
| $36$ | $6$ | $(1,4)(2,5)(3,7)(6,8,9)$ | $2$ |
| $36$ | $6$ | $(4,6,5,8,7,9)$ | $-1$ |
| $36$ | $6$ | $(4,5)(6,8,9)$ | $-2$ |
| $36$ | $6$ | $(1,2,3)(4,5)(6,8,9)$ | $1$ |
| $54$ | $6$ | $(1,2)(4,5)(6,9,8)$ | $0$ |
| $72$ | $6$ | $(1,5,2,7,3,4)(6,8,9)$ | $-1$ |
| $108$ | $6$ | $(1,6,2,8,3,9)(4,5)$ | $-1$ |
| $216$ | $6$ | $(1,6,4,2,8,5)(3,9,7)$ | $0$ |
| $144$ | $9$ | $(1,6,5,2,8,7,3,9,4)$ | $0$ |
| $108$ | $12$ | $(1,4,2,5)(3,7)(6,8,9)$ | $0$ |