Basic invariants
| Dimension: | $6$ |
| Group: | $C_3^2 : D_{6} $ |
| Conductor: | \(8954912000\)\(\medspace = 2^{8} \cdot 5^{3} \cdot 23^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 9.1.18948593792000.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 18T51 |
| Parity: | odd |
| Determinant: | 1.20.2t1.a.a |
| Projective image: | $C_3^2:D_6$ |
| Projective stem field: | Galois closure of 9.1.18948593792000.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 6x^{7} + 12x^{5} - 46x^{4} + 84x^{3} - 23x + 46 \)
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The roots of $f$ are computed in an extension of $\Q_{ 181 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 181 }$:
\( x^{3} + 6x + 179 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 102 + 164\cdot 181 + 113\cdot 181^{2} + 17\cdot 181^{3} + 150\cdot 181^{4} + 128\cdot 181^{5} + 13\cdot 181^{6} + 162\cdot 181^{7} + 48\cdot 181^{8} + 138\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 149 + 161\cdot 181 + 94\cdot 181^{2} + 16\cdot 181^{3} + 96\cdot 181^{4} + 94\cdot 181^{5} + 114\cdot 181^{6} + 42\cdot 181^{7} + 61\cdot 181^{8} + 45\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 171 + 82\cdot 181 + 158\cdot 181^{2} + 161\cdot 181^{3} + 45\cdot 181^{4} + 147\cdot 181^{5} + 94\cdot 181^{6} + 44\cdot 181^{7} + 17\cdot 181^{8} + 45\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 28 a^{2} + 159 a + 35 + \left(143 a^{2} + 151 a + 167\right)\cdot 181 + \left(99 a^{2} + 143 a + 33\right)\cdot 181^{2} + \left(64 a^{2} + 85 a + 36\right)\cdot 181^{3} + \left(143 a^{2} + 54 a + 139\right)\cdot 181^{4} + \left(34 a^{2} + 71 a + 27\right)\cdot 181^{5} + \left(137 a^{2} + 11 a + 82\right)\cdot 181^{6} + \left(17 a^{2} + 156 a + 23\right)\cdot 181^{7} + \left(33 a^{2} + 166 a + 22\right)\cdot 181^{8} + \left(126 a^{2} + 49 a + 162\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 64 a^{2} + 4 a + 179 + \left(18 a^{2} + 19 a + 29\right)\cdot 181 + \left(178 a^{2} + 53 a + 166\right)\cdot 181^{2} + \left(167 a^{2} + 179 a + 87\right)\cdot 181^{3} + \left(74 a^{2} + 13 a + 46\right)\cdot 181^{4} + \left(137 a^{2} + 31 a + 76\right)\cdot 181^{5} + \left(155 a^{2} + 45 a + 156\right)\cdot 181^{6} + \left(117 a^{2} + 121 a + 61\right)\cdot 181^{7} + \left(156 a^{2} + 125 a + 154\right)\cdot 181^{8} + \left(50 a^{2} + 130 a + 41\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 89 a^{2} + 18 a + 98 + \left(19 a^{2} + 10 a + 34\right)\cdot 181 + \left(84 a^{2} + 165 a + 152\right)\cdot 181^{2} + \left(129 a^{2} + 96 a + 114\right)\cdot 181^{3} + \left(143 a^{2} + 112 a + 140\right)\cdot 181^{4} + \left(8 a^{2} + 78 a + 104\right)\cdot 181^{5} + \left(69 a^{2} + 124 a + 171\right)\cdot 181^{6} + \left(45 a^{2} + 84 a + 133\right)\cdot 181^{7} + \left(172 a^{2} + 69 a + 35\right)\cdot 181^{8} + \left(3 a^{2} + 35\right)\cdot 181^{9} +O(181^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 97 a^{2} + 85 a + 83 + \left(79 a^{2} + 164 a + 104\right)\cdot 181 + \left(18 a + 63\right)\cdot 181^{2} + \left(178 a^{2} + 136 a + 84\right)\cdot 181^{3} + \left(14 a^{2} + 111 a + 155\right)\cdot 181^{4} + \left(145 a^{2} + 33 a + 145\right)\cdot 181^{5} + \left(114 a^{2} + 68 a + 6\right)\cdot 181^{6} + \left(29 a^{2} + 17 a + 83\right)\cdot 181^{7} + \left(67 a^{2} + 128 a + 155\right)\cdot 181^{8} + \left(71 a^{2} + 82 a + 8\right)\cdot 181^{9} +O(181^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 106 a^{2} + 123 a + 119 + \left(90 a^{2} + 9 a + 148\right)\cdot 181 + \left(179 a^{2} + 19 a + 55\right)\cdot 181^{2} + \left(45 a^{2} + 4 a + 99\right)\cdot 181^{3} + \left(82 a^{2} + 24 a + 62\right)\cdot 181^{4} + \left(148 a^{2} + 63 a + 159\right)\cdot 181^{5} + \left(108 a^{2} + 136 a + 163\right)\cdot 181^{6} + \left(106 a^{2} + 59 a + 28\right)\cdot 181^{7} + \left(93 a^{2} + 84 a + 80\right)\cdot 181^{8} + \left(4 a^{2} + 157 a + 103\right)\cdot 181^{9} +O(181^{10})\)
|
| $r_{ 9 }$ | $=$ |
\( 159 a^{2} + 154 a + 150 + \left(10 a^{2} + 6 a + 10\right)\cdot 181 + \left(a^{2} + 143 a + 66\right)\cdot 181^{2} + \left(138 a^{2} + 40 a + 105\right)\cdot 181^{3} + \left(83 a^{2} + 45 a + 68\right)\cdot 181^{4} + \left(68 a^{2} + 84 a + 20\right)\cdot 181^{5} + \left(138 a^{2} + 157 a + 101\right)\cdot 181^{6} + \left(44 a^{2} + 103 a + 143\right)\cdot 181^{7} + \left(20 a^{2} + 149 a + 148\right)\cdot 181^{8} + \left(105 a^{2} + 121 a + 143\right)\cdot 181^{9} +O(181^{10})\)
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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 |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(4,5)(7,8)$ | $0$ |
| $9$ | $2$ | $(1,6)(2,5)(3,4)$ | $-2$ |
| $9$ | $2$ | $(1,6)(2,4)(3,5)(7,8)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,6,5)(7,9,8)$ | $-3$ |
| $6$ | $3$ | $(1,6,9)(2,5,8)(3,4,7)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(7,9,8)$ | $0$ |
| $12$ | $3$ | $(1,5,8)(2,4,7)(3,6,9)$ | $0$ |
| $18$ | $6$ | $(1,6,9)(2,4,8,3,5,7)$ | $0$ |
| $18$ | $6$ | $(1,5,3,6,2,4)(7,9,8)$ | $1$ |
| $18$ | $6$ | $(1,9,3,8,2,7)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.