Basic invariants
| Dimension: | $3$ |
| Group: | $C_3 \wr S_3 $ |
| Conductor: | \(20991479\)\(\medspace = 23 \cdot 97^{3} \) |
| Artin stem field: | Galois closure of 9.3.114479303.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_3 \wr S_3 $ |
| Parity: | odd |
| Determinant: | 1.2231.6t1.a.a |
| Projective image: | $C_3^2:C_6$ |
| Projective stem field: | Galois closure of 9.1.24774122524321.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{9} - 3x^{8} + 4x^{7} - x^{6} - 7x^{5} + 11x^{4} - 9x^{3} + 3x^{2} - x + 1 \)
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The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$:
\( x^{3} + 2x + 209 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 58 + 93\cdot 211 + 140\cdot 211^{2} + 177\cdot 211^{3} + 174\cdot 211^{4} + 63\cdot 211^{5} + 107\cdot 211^{6} + 69\cdot 211^{7} + 182\cdot 211^{8} + 171\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 162 + 192\cdot 211 + 38\cdot 211^{2} + 52\cdot 211^{3} + 180\cdot 211^{4} + 34\cdot 211^{5} + 134\cdot 211^{6} + 156\cdot 211^{7} + 185\cdot 211^{8} + 179\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 203 + 135\cdot 211 + 31\cdot 211^{2} + 192\cdot 211^{3} + 66\cdot 211^{4} + 112\cdot 211^{5} + 180\cdot 211^{6} + 195\cdot 211^{7} + 53\cdot 211^{8} + 70\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 a^{2} + 40 a + 193 + \left(56 a^{2} + 19 a + 74\right)\cdot 211 + \left(44 a^{2} + 143 a + 129\right)\cdot 211^{2} + \left(18 a^{2} + 34 a + 94\right)\cdot 211^{3} + \left(86 a^{2} + 176 a + 44\right)\cdot 211^{4} + \left(94 a^{2} + 100 a + 196\right)\cdot 211^{5} + \left(96 a^{2} + 14 a + 198\right)\cdot 211^{6} + \left(12 a^{2} + 195 a + 86\right)\cdot 211^{7} + \left(33 a^{2} + 13 a + 114\right)\cdot 211^{8} + \left(159 a^{2} + 205 a + 71\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 65 a^{2} + 32 a + 87 + \left(110 a^{2} + 94 a + 6\right)\cdot 211 + \left(210 a^{2} + 57 a + 140\right)\cdot 211^{2} + \left(162 a^{2} + 198 a + 76\right)\cdot 211^{3} + \left(4 a^{2} + 41 a + 6\right)\cdot 211^{4} + \left(142 a^{2} + 14 a + 119\right)\cdot 211^{5} + \left(166 a^{2} + 147 a + 81\right)\cdot 211^{6} + \left(62 a^{2} + 104 a + 13\right)\cdot 211^{7} + \left(92 a^{2} + 19 a + 123\right)\cdot 211^{8} + \left(14 a^{2} + 117 a + 89\right)\cdot 211^{9} +O(211^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 103 a^{2} + 184 a + 208 + \left(70 a^{2} + 60 a + 93\right)\cdot 211 + \left(27 a^{2} + 125 a + 36\right)\cdot 211^{2} + \left(25 a^{2} + 2 a + 174\right)\cdot 211^{3} + \left(8 a^{2} + 196 a + 10\right)\cdot 211^{4} + \left(87 a^{2} + 138 a + 116\right)\cdot 211^{5} + \left(166 a^{2} + 11 a + 151\right)\cdot 211^{6} + \left(56 a^{2} + 98 a + 75\right)\cdot 211^{7} + \left(207 a^{2} + 197 a + 65\right)\cdot 211^{8} + \left(15 a^{2} + 40 a + 21\right)\cdot 211^{9} +O(211^{10})\)
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| $r_{ 7 }$ | $=$ |
\( 107 a^{2} + 139 a + 143 + \left(44 a^{2} + 97 a + 129\right)\cdot 211 + \left(167 a^{2} + 10 a + 152\right)\cdot 211^{2} + \left(29 a^{2} + 189 a + 39\right)\cdot 211^{3} + \left(120 a^{2} + 203 a + 160\right)\cdot 211^{4} + \left(185 a^{2} + 95 a + 106\right)\cdot 211^{5} + \left(158 a^{2} + 49 a + 141\right)\cdot 211^{6} + \left(135 a^{2} + 122 a + 110\right)\cdot 211^{7} + \left(85 a^{2} + 177 a + 184\right)\cdot 211^{8} + \left(37 a^{2} + 99 a + 49\right)\cdot 211^{9} +O(211^{10})\)
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| $r_{ 8 }$ | $=$ |
\( 116 a^{2} + 208 a + 155 + \left(165 a^{2} + 86 a + 9\right)\cdot 211 + \left(184 a^{2} + 133 a + 176\right)\cdot 211^{2} + \left(35 a^{2} + 6 a + 47\right)\cdot 211^{3} + \left(102 a^{2} + 55 a + 136\right)\cdot 211^{4} + \left(205 a^{2} + 67 a + 203\right)\cdot 211^{5} + \left(133 a + 141\right)\cdot 211^{6} + \left(9 a^{2} + 197 a + 152\right)\cdot 211^{7} + \left(11 a^{2} + 27 a + 14\right)\cdot 211^{8} + \left(177 a^{2} + 182 a + 25\right)\cdot 211^{9} +O(211^{10})\)
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| $r_{ 9 }$ | $=$ |
\( 203 a^{2} + 30 a + 60 + \left(185 a^{2} + 63 a + 107\right)\cdot 211 + \left(209 a^{2} + 163 a + 209\right)\cdot 211^{2} + \left(149 a^{2} + 201 a + 199\right)\cdot 211^{3} + \left(100 a^{2} + 170 a + 63\right)\cdot 211^{4} + \left(129 a^{2} + 4 a + 102\right)\cdot 211^{5} + \left(43 a^{2} + 66 a + 128\right)\cdot 211^{6} + \left(145 a^{2} + 126 a + 193\right)\cdot 211^{7} + \left(203 a^{2} + 196 a + 130\right)\cdot 211^{8} + \left(17 a^{2} + 198 a + 164\right)\cdot 211^{9} +O(211^{10})\)
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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 | Complex conjugation |
| $1$ | $1$ | $()$ | $3$ | |
| $9$ | $2$ | $(1,6)(2,8)(3,9)$ | $1$ | ✓ |
| $1$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $-3 \zeta_{3} - 3$ | |
| $1$ | $3$ | $(1,3,2)(4,7,5)(6,9,8)$ | $3 \zeta_{3}$ | |
| $3$ | $3$ | $(1,2,3)$ | $-\zeta_{3} - 2$ | |
| $3$ | $3$ | $(1,3,2)$ | $\zeta_{3} - 1$ | |
| $3$ | $3$ | $(1,3,2)(4,5,7)(6,9,8)$ | $-2 \zeta_{3} - 1$ | |
| $3$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $2 \zeta_{3} + 1$ | |
| $3$ | $3$ | $(1,2,3)(4,5,7)$ | $\zeta_{3} + 2$ | |
| $3$ | $3$ | $(1,3,2)(4,7,5)$ | $-\zeta_{3} + 1$ | |
| $6$ | $3$ | $(1,3,2)(4,5,7)$ | $0$ | |
| $18$ | $3$ | $(1,7,6)(2,4,8)(3,5,9)$ | $0$ | |
| $9$ | $6$ | $(1,6)(2,8)(3,9)(4,7,5)$ | $-\zeta_{3} - 1$ | |
| $9$ | $6$ | $(1,6)(2,8)(3,9)(4,5,7)$ | $\zeta_{3}$ | |
| $9$ | $6$ | $(1,8,2,9,3,6)(4,7,5)$ | $\zeta_{3}$ | |
| $9$ | $6$ | $(1,6,3,9,2,8)(4,5,7)$ | $-\zeta_{3} - 1$ | |
| $9$ | $6$ | $(1,9,3,8,2,6)(4,7,5)$ | $1$ | |
| $9$ | $6$ | $(1,6,2,8,3,9)(4,5,7)$ | $1$ | |
| $9$ | $6$ | $(1,4,2,5,3,7)$ | $-\zeta_{3} - 1$ | |
| $9$ | $6$ | $(1,7,3,5,2,4)$ | $\zeta_{3}$ | |
| $18$ | $9$ | $(1,5,9,3,4,8,2,7,6)$ | $0$ | |
| $18$ | $9$ | $(1,9,4,2,6,5,3,8,7)$ | $0$ |