Basic invariants
| Dimension: | $12$ |
| Group: | $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ |
| Conductor: | \(7109802703089303552\)\(\medspace = 2^{22} \cdot 3^{5} \cdot 17^{8} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.3.166838876928.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T219 |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_3^3:S_4$ |
| Projective stem field: | Galois closure of 9.3.166838876928.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} + x^{7} - 5x^{6} + 6x^{5} - 9x^{4} - 8x^{3} - 10x^{2} + 16x - 4 \)
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The roots of $f$ are computed in an extension of $\Q_{ 199 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 199 }$:
\( x^{3} + x + 196 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 50\cdot 199 + 124\cdot 199^{2} + 17\cdot 199^{3} + 73\cdot 199^{4} + 67\cdot 199^{5} + 32\cdot 199^{6} + 34\cdot 199^{7} + 59\cdot 199^{8} + 64\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 66\cdot 199 + 108\cdot 199^{2} + 83\cdot 199^{3} + 77\cdot 199^{4} + 77\cdot 199^{5} + 74\cdot 199^{6} + 106\cdot 199^{7} + 16\cdot 199^{8} + 80\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 127 + 111\cdot 199 + 114\cdot 199^{2} + 22\cdot 199^{3} + 10\cdot 199^{4} + 159\cdot 199^{5} + 155\cdot 199^{6} + 2\cdot 199^{7} + 81\cdot 199^{8} + 67\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 7 a^{2} + 79 a + 84 + \left(43 a^{2} + 79 a + 7\right)\cdot 199 + \left(147 a^{2} + 84 a + 83\right)\cdot 199^{2} + \left(42 a^{2} + 78 a + 128\right)\cdot 199^{3} + \left(184 a^{2} + 178 a + 189\right)\cdot 199^{4} + \left(57 a^{2} + 195 a + 129\right)\cdot 199^{5} + \left(112 a^{2} + 119 a + 28\right)\cdot 199^{6} + \left(6 a^{2} + 179 a + 139\right)\cdot 199^{7} + \left(188 a^{2} + 115 a + 115\right)\cdot 199^{8} + \left(104 a^{2} + 184 a + 136\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 79 a^{2} + 184 a + 132 + \left(142 a^{2} + 40 a + 73\right)\cdot 199 + \left(96 a^{2} + 79 a + 49\right)\cdot 199^{2} + \left(110 a^{2} + 147 a + 107\right)\cdot 199^{3} + \left(102 a^{2} + 31 a + 2\right)\cdot 199^{4} + \left(169 a^{2} + 38 a + 138\right)\cdot 199^{5} + \left(151 a^{2} + 59 a + 187\right)\cdot 199^{6} + \left(181 a^{2} + 107 a + 56\right)\cdot 199^{7} + \left(57 a^{2} + 182 a + 95\right)\cdot 199^{8} + \left(101 a^{2} + 122 a + 1\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 90 a^{2} + 173 a + 62 + \left(154 a^{2} + 49 a + 48\right)\cdot 199 + \left(12 a^{2} + 115 a + 173\right)\cdot 199^{2} + \left(193 a^{2} + 6 a + 53\right)\cdot 199^{3} + \left(4 a^{2} + 52 a + 148\right)\cdot 199^{4} + \left(110 a^{2} + 135 a + 79\right)\cdot 199^{5} + \left(81 a^{2} + 108 a + 79\right)\cdot 199^{6} + \left(99 a^{2} + 188 a + 16\right)\cdot 199^{7} + \left(150 a^{2} + 75 a + 124\right)\cdot 199^{8} + \left(85 a^{2} + 66 a + 52\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 113 a^{2} + 135 a + 22 + \left(13 a^{2} + 78 a + 54\right)\cdot 199 + \left(154 a^{2} + 35 a + 21\right)\cdot 199^{2} + \left(45 a^{2} + 172 a + 64\right)\cdot 199^{3} + \left(111 a^{2} + 187 a + 8\right)\cdot 199^{4} + \left(170 a^{2} + 163 a + 6\right)\cdot 199^{5} + \left(133 a^{2} + 19 a + 43\right)\cdot 199^{6} + \left(10 a^{2} + 111 a + 9\right)\cdot 199^{7} + \left(152 a^{2} + 99 a + 158\right)\cdot 199^{8} + \left(191 a^{2} + 90 a + 61\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 138 a^{2} + 7 a + 94 + \left(97 a^{2} + 125 a + 10\right)\cdot 199 + \left(62 a^{2} + 166 a + 140\right)\cdot 199^{2} + \left(38 a^{2} + 89 a + 149\right)\cdot 199^{3} + \left(135 a^{2} + 6 a + 168\right)\cdot 199^{4} + \left(21 a^{2} + 4 a + 20\right)\cdot 199^{5} + \left(194 a^{2} + 179 a + 88\right)\cdot 199^{6} + \left(168 a^{2} + 182 a + 195\right)\cdot 199^{7} + \left(7 a^{2} + 100 a + 28\right)\cdot 199^{8} + \left(162 a^{2} + 67 a + 37\right)\cdot 199^{9} +O(199^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 170 a^{2} + 19 a + 49 + \left(145 a^{2} + 24 a + 175\right)\cdot 199 + \left(123 a^{2} + 116 a + 180\right)\cdot 199^{2} + \left(166 a^{2} + 102 a + 168\right)\cdot 199^{3} + \left(58 a^{2} + 140 a + 117\right)\cdot 199^{4} + \left(67 a^{2} + 59 a + 117\right)\cdot 199^{5} + \left(122 a^{2} + 110 a + 106\right)\cdot 199^{6} + \left(129 a^{2} + 26 a + 36\right)\cdot 199^{7} + \left(40 a^{2} + 22 a + 117\right)\cdot 199^{8} + \left(150 a^{2} + 65 a + 95\right)\cdot 199^{9} +O(199^{10})\)
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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 |
| $1$ | $1$ | $()$ | $12$ |
| $18$ | $2$ | $(1,9)(2,6)(3,8)$ | $2$ |
| $27$ | $2$ | $(1,3)(5,7)$ | $0$ |
| $4$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $3$ |
| $4$ | $3$ | $(1,3,2)(4,5,7)(6,9,8)$ | $3$ |
| $6$ | $3$ | $(6,8,9)$ | $0$ |
| $12$ | $3$ | $(1,2,3)(4,7,5)$ | $-3$ |
| $72$ | $3$ | $(1,5,9)(2,7,6)(3,4,8)$ | $0$ |
| $162$ | $4$ | $(1,9,3,6)(2,8)(4,7)$ | $0$ |
| $18$ | $6$ | $(1,5)(2,7)(3,4)(6,9,8)$ | $2$ |
| $18$ | $6$ | $(1,5)(2,7)(3,4)(6,8,9)$ | $2$ |
| $36$ | $6$ | $(1,8,2,9,3,6)(4,5,7)$ | $-1$ |
| $36$ | $6$ | $(1,6,3,9,2,8)(4,7,5)$ | $-1$ |
| $36$ | $6$ | $(1,7,2,5,3,4)$ | $-1$ |
| $54$ | $6$ | $(1,3)(5,7)(6,9,8)$ | $0$ |
| $72$ | $9$ | $(1,7,6,2,5,8,3,4,9)$ | $0$ |
| $72$ | $9$ | $(1,6,5,3,9,7,2,8,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.