Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 8.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{3} + 2 x + 209 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 24 + 210\cdot 211 + 55\cdot 211^{2} + 184\cdot 211^{3} + 51\cdot 211^{4} + 126\cdot 211^{5} + 7\cdot 211^{6} + 183\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 + 203\cdot 211 + 61\cdot 211^{2} + 126\cdot 211^{3} + 7\cdot 211^{4} + 51\cdot 211^{5} + 23\cdot 211^{6} + 199\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 125 + 8\cdot 211 + 93\cdot 211^{2} + 111\cdot 211^{3} + 151\cdot 211^{4} + 33\cdot 211^{5} + 180\cdot 211^{6} + 39\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a^{2} + 158 a + 76 + \left(35 a^{2} + 31 a + 117\right)\cdot 211 + \left(133 a^{2} + 85 a + 177\right)\cdot 211^{2} + \left(90 a^{2} + 151 a + 120\right)\cdot 211^{3} + \left(148 a^{2} + 32 a + 127\right)\cdot 211^{4} + \left(42 a^{2} + 14 a + 197\right)\cdot 211^{5} + \left(128 a^{2} + 150 a + 170\right)\cdot 211^{6} + \left(16 a^{2} + 189 a + 162\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 75 a^{2} + 6 a + 100 + \left(110 a^{2} + 60 a + 6\right)\cdot 211 + \left(11 a^{2} + 66 a + 156\right)\cdot 211^{2} + \left(142 a^{2} + 77 a + 48\right)\cdot 211^{3} + \left(22 a^{2} + 175 a + 30\right)\cdot 211^{4} + \left(77 a^{2} + 190 a + 173\right)\cdot 211^{5} + \left(83 a^{2} + 197 a + 40\right)\cdot 211^{6} + \left(190 a^{2} + 121 a + 113\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 141 a^{2} + 121 a + 188 + \left(151 a^{2} + 96 a + 131\right)\cdot 211 + \left(4 a^{2} + 55 a + 76\right)\cdot 211^{2} + \left(82 a^{2} + 110 a + 109\right)\cdot 211^{3} + \left(51 a^{2} + 153 a + 68\right)\cdot 211^{4} + \left(6 a^{2} + 181 a + 8\right)\cdot 211^{5} + \left(165 a^{2} + 33 a + 9\right)\cdot 211^{6} + \left(22 a^{2} + 53 a + 171\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 165 a^{2} + 102 a + 9 + \left(200 a^{2} + 2 a + 127\right)\cdot 211 + \left(153 a^{2} + 2 a + 64\right)\cdot 211^{2} + \left(156 a^{2} + 39 a + 68\right)\cdot 211^{3} + \left(90 a^{2} + 90 a + 191\right)\cdot 211^{4} + \left(84 a^{2} + 47 a + 182\right)\cdot 211^{5} + \left(89 a^{2} + 203 a + 48\right)\cdot 211^{6} + \left(107 a^{2} + 184 a + 143\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 200 a^{2} + 162 a + 126 + \left(185 a^{2} + 176 a + 177\right)\cdot 211 + \left(134 a^{2} + 123 a + 179\right)\cdot 211^{2} + \left(174 a^{2} + 20 a + 21\right)\cdot 211^{3} + \left(182 a^{2} + 88 a + 103\right)\cdot 211^{4} + \left(83 a^{2} + 149 a + 41\right)\cdot 211^{5} + \left(204 a^{2} + 68 a + 202\right)\cdot 211^{6} + \left(86 a^{2} + 47 a + 115\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 206 a^{2} + 84 a + 134 + \left(159 a^{2} + 54 a + 72\right)\cdot 211 + \left(194 a^{2} + 89 a + 189\right)\cdot 211^{2} + \left(197 a^{2} + 23 a + 52\right)\cdot 211^{3} + \left(136 a^{2} + 93 a + 112\right)\cdot 211^{4} + \left(127 a^{2} + 49 a + 29\right)\cdot 211^{5} + \left(173 a^{2} + 190 a + 161\right)\cdot 211^{6} + \left(208 a^{2} + 35 a + 137\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,8,7)(5,6,9)$ |
| $(2,3)(4,7)(5,6)$ |
| $(1,7,5)(2,4,6)(3,8,9)$ |
| $(4,7,8)(5,6,9)$ |
| $(4,6)(5,7)(8,9)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(6,9)(7,8)$ | $0$ |
| $9$ | $2$ | $(4,6)(5,7)(8,9)$ | $2$ |
| $9$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,8,7)(5,6,9)$ | $-3$ |
| $6$ | $3$ | $(1,5,4)(2,6,8)(3,9,7)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(4,8,7)$ | $0$ |
| $12$ | $3$ | $(1,7,5)(2,4,6)(3,8,9)$ | $0$ |
| $18$ | $6$ | $(1,4,5)(2,7,6,3,8,9)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,9,7,6,8,5)$ | $-1$ |
| $18$ | $6$ | $(1,4,3,8,2,7)(5,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
