Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 9.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{3} + 2 x + 209 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 49 + 198\cdot 211 + 10\cdot 211^{2} + 177\cdot 211^{3} + 101\cdot 211^{4} + 56\cdot 211^{5} + 81\cdot 211^{6} + 64\cdot 211^{7} + 7\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 104 + 28\cdot 211 + 158\cdot 211^{2} + 154\cdot 211^{3} + 43\cdot 211^{4} + 19\cdot 211^{5} + 54\cdot 211^{6} + 16\cdot 211^{7} + 92\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 141 + 106\cdot 211 + 134\cdot 211^{2} + 118\cdot 211^{3} + 25\cdot 211^{4} + 176\cdot 211^{5} + 106\cdot 211^{6} + 207\cdot 211^{7} + 126\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 a^{2} + 44 a + 109 + \left(157 a^{2} + 4 a + 142\right)\cdot 211 + \left(72 a^{2} + 168 a + 24\right)\cdot 211^{2} + \left(207 a^{2} + 160 a + 25\right)\cdot 211^{3} + \left(99 a^{2} + 7 a + 116\right)\cdot 211^{4} + \left(111 a^{2} + 26 a + 202\right)\cdot 211^{5} + \left(2 a^{2} + 157 a + 160\right)\cdot 211^{6} + \left(18 a^{2} + 193 a + 187\right)\cdot 211^{7} + \left(109 a^{2} + 8 a + 50\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 a^{2} + 161 a + 198 + \left(67 a^{2} + 164 a + 185\right)\cdot 211 + \left(61 a^{2} + 84 a + 52\right)\cdot 211^{2} + \left(181 a^{2} + 32 a + 202\right)\cdot 211^{3} + \left(16 a^{2} + 183 a + 52\right)\cdot 211^{4} + \left(6 a^{2} + 63 a + 81\right)\cdot 211^{5} + \left(176 a^{2} + 120 a + 207\right)\cdot 211^{6} + \left(177 a^{2} + 86 a + 117\right)\cdot 211^{7} + \left(4 a^{2} + 153 a + 25\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 49 a^{2} + 122 a + 23 + \left(104 a^{2} + 132 a + 165\right)\cdot 211 + \left(37 a^{2} + 114 a + 161\right)\cdot 211^{2} + \left(127 a^{2} + 90 a + 59\right)\cdot 211^{3} + \left(76 a^{2} + 190 a + 62\right)\cdot 211^{4} + \left(193 a^{2} + 150 a + 190\right)\cdot 211^{5} + \left(192 a^{2} + 72 a + 18\right)\cdot 211^{6} + \left(158 a^{2} + 22 a + 163\right)\cdot 211^{7} + \left(60 a^{2} + 127 a + 29\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 76 a^{2} + 67 a + 47 + \left(49 a^{2} + 66 a + 69\right)\cdot 211 + \left(144 a^{2} + 26 a + 190\right)\cdot 211^{2} + \left(131 a^{2} + 16 a + 64\right)\cdot 211^{3} + \left(17 a^{2} + 55 a + 6\right)\cdot 211^{4} + \left(29 a^{2} + 76 a + 163\right)\cdot 211^{5} + \left(210 a^{2} + 39 a + 15\right)\cdot 211^{6} + \left(67 a^{2} + 91 a + 184\right)\cdot 211^{7} + \left(142 a^{2} + 188 a + 24\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 118 a^{2} + 100 a + 103 + \left(4 a^{2} + 140 a + 9\right)\cdot 211 + \left(205 a^{2} + 16 a + 201\right)\cdot 211^{2} + \left(82 a^{2} + 34 a + 210\right)\cdot 211^{3} + \left(93 a^{2} + 148 a + 36\right)\cdot 211^{4} + \left(70 a^{2} + 108 a + 7\right)\cdot 211^{5} + \left(209 a^{2} + 14 a + 85\right)\cdot 211^{6} + \left(124 a^{2} + 137 a + 119\right)\cdot 211^{7} + \left(170 a^{2} + 13 a + 62\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 140 a^{2} + 139 a + 74 + \left(39 a^{2} + 124 a + 149\right)\cdot 211 + \left(112 a^{2} + 11 a + 120\right)\cdot 211^{2} + \left(113 a^{2} + 88 a + 41\right)\cdot 211^{3} + \left(117 a^{2} + 48 a + 187\right)\cdot 211^{4} + \left(11 a^{2} + 207 a + 158\right)\cdot 211^{5} + \left(53 a^{2} + 17 a + 113\right)\cdot 211^{6} + \left(85 a^{2} + 102 a + 205\right)\cdot 211^{7} + \left(145 a^{2} + 141 a + 1\right)\cdot 211^{8} +O\left(211^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,8,7)(5,6,9)$ |
| $(1,3)(4,7)(5,6)$ |
| $(4,7,8)(5,6,9)$ |
| $(4,6)(5,7)(8,9)$ |
| $(1,8,9)(2,7,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,3)(6,9)(7,8)$ |
$0$ |
| $9$ |
$2$ |
$(4,6)(5,7)(8,9)$ |
$-2$ |
| $9$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$3$ |
$(1,2,3)(4,8,7)(5,6,9)$ |
$-3$ |
| $6$ |
$3$ |
$(1,9,7)(2,5,4)(3,6,8)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(4,8,7)$ |
$0$ |
| $12$ |
$3$ |
$(1,8,9)(2,7,5)(3,4,6)$ |
$0$ |
| $18$ |
$6$ |
$(1,8,9,3,7,6)(2,4,5)$ |
$0$ |
| $18$ |
$6$ |
$(1,2,3)(4,9,7,6,8,5)$ |
$1$ |
| $18$ |
$6$ |
$(1,8,3,7,2,4)(5,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
