Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(335544320000000000\)\(\medspace = 2^{35} \cdot 5^{10} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.25600000000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T218 |
| Parity: | odd |
| Determinant: | 1.8.2t1.b.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.1.25600000000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 2x^{8} + 4x^{7} - 8x^{6} + 21x^{5} - 38x^{4} + 46x^{3} - 32x^{2} + 24x - 8 \)
|
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:
| $r_{ 1 }$ | $=$ |
\( 98 + 215\cdot 227 + 132\cdot 227^{2} + 20\cdot 227^{3} + 126\cdot 227^{4} + 108\cdot 227^{5} + 213\cdot 227^{6} + 44\cdot 227^{7} + 207\cdot 227^{8} + 224\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 116 + 190\cdot 227 + 194\cdot 227^{2} + 44\cdot 227^{3} + 181\cdot 227^{4} + 78\cdot 227^{5} + 113\cdot 227^{6} + 194\cdot 227^{7} + 6\cdot 227^{8} + 70\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 146 + 67\cdot 227 + 112\cdot 227^{2} + 4\cdot 227^{3} + 68\cdot 227^{4} + 21\cdot 227^{5} + 161\cdot 227^{6} + 7\cdot 227^{7} + 115\cdot 227^{8} + 30\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 72 a^{2} + 148 a + 50 + \left(188 a^{2} + 16 a + 212\right)\cdot 227 + \left(29 a^{2} + 176 a + 157\right)\cdot 227^{2} + \left(190 a^{2} + 115 a + 93\right)\cdot 227^{3} + \left(48 a^{2} + 199 a + 104\right)\cdot 227^{4} + \left(8 a^{2} + 221 a + 218\right)\cdot 227^{5} + \left(186 a^{2} + 115 a + 83\right)\cdot 227^{6} + \left(211 a^{2} + 6 a + 200\right)\cdot 227^{7} + \left(148 a^{2} + 225 a + 64\right)\cdot 227^{8} + \left(90 a^{2} + 46 a + 179\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 96 a^{2} + 123 a + 206 + \left(49 a^{2} + 166 a + 173\right)\cdot 227 + \left(174 a^{2} + 78 a + 118\right)\cdot 227^{2} + \left(120 a^{2} + 144 a + 70\right)\cdot 227^{3} + \left(150 a^{2} + 137 a + 36\right)\cdot 227^{4} + \left(100 a^{2} + 214 a + 84\right)\cdot 227^{5} + \left(57 a^{2} + 96 a + 2\right)\cdot 227^{6} + \left(133 a^{2} + 18 a + 26\right)\cdot 227^{7} + \left(31 a^{2} + 150 a + 66\right)\cdot 227^{8} + \left(183 a^{2} + 49 a + 77\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 140 a^{2} + 173 a + 189 + \left(174 a^{2} + 154 a + 113\right)\cdot 227 + \left(132 a^{2} + 171 a + 63\right)\cdot 227^{2} + \left(221 a^{2} + 155 a + 129\right)\cdot 227^{3} + \left(201 a^{2} + 147 a + 180\right)\cdot 227^{4} + \left(31 a^{2} + 149 a + 143\right)\cdot 227^{5} + \left(226 a^{2} + 52 a + 151\right)\cdot 227^{6} + \left(63 a^{2} + 69 a + 160\right)\cdot 227^{7} + \left(44 a^{2} + 21 a + 158\right)\cdot 227^{8} + \left(165 a^{2} + 24 a + 204\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 168 a^{2} + 168 a + 178 + \left(146 a^{2} + 76 a + 156\right)\cdot 227 + \left(120 a^{2} + 216 a + 127\right)\cdot 227^{2} + \left(134 a^{2} + 11 a + 19\right)\cdot 227^{3} + \left(166 a^{2} + 42 a + 110\right)\cdot 227^{4} + \left(124 a^{2} + 195 a + 222\right)\cdot 227^{5} + \left(162 a^{2} + 47 a + 203\right)\cdot 227^{6} + \left(66 a^{2} + 16 a + 6\right)\cdot 227^{7} + \left(41 a^{2} + 5 a + 224\right)\cdot 227^{8} + \left(43 a^{2} + 83 a + 115\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 214 a^{2} + 138 a + 88 + \left(118 a^{2} + 133 a + 195\right)\cdot 227 + \left(76 a^{2} + 61 a + 68\right)\cdot 227^{2} + \left(129 a^{2} + 99 a + 88\right)\cdot 227^{3} + \left(11 a^{2} + 212 a + 130\right)\cdot 227^{4} + \left(94 a^{2} + 36 a + 181\right)\cdot 227^{5} + \left(105 a^{2} + 63 a + 127\right)\cdot 227^{6} + \left(175 a^{2} + 204 a\right)\cdot 227^{7} + \left(36 a^{2} + 223 a + 218\right)\cdot 227^{8} + \left(93 a^{2} + 96 a + 106\right)\cdot 227^{9} +O(227^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 218 a^{2} + 158 a + 66 + \left(2 a^{2} + 132 a + 36\right)\cdot 227 + \left(147 a^{2} + 203 a + 158\right)\cdot 227^{2} + \left(111 a^{2} + 153 a + 209\right)\cdot 227^{3} + \left(101 a^{2} + 168 a + 197\right)\cdot 227^{4} + \left(94 a^{2} + 89 a + 75\right)\cdot 227^{5} + \left(170 a^{2} + 77 a + 77\right)\cdot 227^{6} + \left(29 a^{2} + 139 a + 39\right)\cdot 227^{7} + \left(151 a^{2} + 55 a + 74\right)\cdot 227^{8} + \left(105 a^{2} + 153 a + 125\right)\cdot 227^{9} +O(227^{10})\)
|
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 value |
| $1$ | $1$ | $()$ | $12$ |
| $27$ | $2$ | $(1,2)(5,6)$ | $0$ |
| $54$ | $2$ | $(1,2)(4,5)(6,7)(8,9)$ | $2$ |
| $6$ | $3$ | $(1,3,2)$ | $0$ |
| $8$ | $3$ | $(1,2,3)(4,7,8)(5,6,9)$ | $3$ |
| $12$ | $3$ | $(1,3,2)(5,9,6)$ | $-3$ |
| $72$ | $3$ | $(1,4,5)(2,7,6)(3,8,9)$ | $0$ |
| $54$ | $4$ | $(1,6,2,5)(3,9)$ | $0$ |
| $54$ | $6$ | $(1,2,3)(4,7)(5,6)$ | $0$ |
| $108$ | $6$ | $(1,9,3,6,2,5)(4,7)$ | $-1$ |
| $72$ | $9$ | $(1,7,6,2,8,9,3,4,5)$ | $0$ |
| $72$ | $9$ | $(1,8,9,3,7,6,2,4,5)$ | $0$ |
| $54$ | $12$ | $(1,3,2)(4,6,7,5)(8,9)$ | $0$ |
| $54$ | $12$ | $(1,2,3)(4,6,7,5)(8,9)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.