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:
| $r_{ 1 }$ |
$=$ |
$ 101 + 14\cdot 211 + 131\cdot 211^{2} + 37\cdot 211^{3} + 59\cdot 211^{4} + 184\cdot 211^{5} + 152\cdot 211^{6} + 167\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 148 + 53\cdot 211 + 208\cdot 211^{2} + 126\cdot 211^{3} + 139\cdot 211^{4} + 9\cdot 211^{5} + 181\cdot 211^{6} + 125\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 173 + 142\cdot 211 + 82\cdot 211^{2} + 46\cdot 211^{3} + 12\cdot 211^{4} + 17\cdot 211^{5} + 88\cdot 211^{6} + 128\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 25 a^{2} + 2 a + 174 + \left(186 a^{2} + a + 177\right)\cdot 211 + \left(60 a^{2} + 8 a + 10\right)\cdot 211^{2} + \left(108 a^{2} + 82 a + 74\right)\cdot 211^{3} + \left(61 a^{2} + 152 a + 152\right)\cdot 211^{4} + \left(143 a^{2} + 15 a + 120\right)\cdot 211^{5} + \left(205 a^{2} + 49 a + 133\right)\cdot 211^{6} + \left(23 a^{2} + 21 a + 172\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 35 a^{2} + 60 a + 117 + \left(204 a^{2} + 37 a + 131\right)\cdot 211 + \left(9 a^{2} + 50 a + 83\right)\cdot 211^{2} + \left(19 a^{2} + 2 a + 25\right)\cdot 211^{3} + \left(138 a^{2} + 32 a + 184\right)\cdot 211^{4} + \left(91 a^{2} + 25 a + 51\right)\cdot 211^{5} + \left(113 a^{2} + 112 a + 151\right)\cdot 211^{6} + \left(141 a^{2} + 172 a + 188\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 47 a^{2} + 181 a + 133 + \left(61 a^{2} + 35 a + 81\right)\cdot 211 + \left(70 a^{2} + 132 a + 23\right)\cdot 211^{2} + \left(72 a^{2} + 145 a + 26\right)\cdot 211^{3} + \left(24 a^{2} + 204 a + 173\right)\cdot 211^{4} + \left(8 a^{2} + 127 a + 10\right)\cdot 211^{5} + \left(201 a^{2} + 151 a + 57\right)\cdot 211^{6} + \left(136 a^{2} + 75 a + 112\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 68 a^{2} + 207 a + 161 + \left(172 a^{2} + 46 a + 18\right)\cdot 211 + \left(200 a^{2} + 25 a + 127\right)\cdot 211^{2} + \left(78 a^{2} + 91 a + 175\right)\cdot 211^{3} + \left(182 a^{2} + 153 a + 172\right)\cdot 211^{4} + \left(72 a^{2} + 150 a + 26\right)\cdot 211^{5} + \left(192 a^{2} + 176 a + 186\right)\cdot 211^{6} + \left(77 a^{2} + 189 a + 103\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 108 a^{2} + 155 a + 144 + \left(45 a^{2} + 126 a + 60\right)\cdot 211 + 135 a\cdot 211^{2} + \left(113 a^{2} + 117 a + 10\right)\cdot 211^{3} + \left(101 a^{2} + 25 a + 65\right)\cdot 211^{4} + \left(46 a^{2} + 35 a + 132\right)\cdot 211^{5} + \left(116 a^{2} + 133 a + 84\right)\cdot 211^{6} + \left(202 a^{2} + 59 a + 129\right)\cdot 211^{7} +O\left(211^{ 8 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 139 a^{2} + 28 a + 115 + \left(174 a^{2} + 174 a + 162\right)\cdot 211 + \left(79 a^{2} + 70 a + 176\right)\cdot 211^{2} + \left(30 a^{2} + 194 a + 110\right)\cdot 211^{3} + \left(125 a^{2} + 64 a + 96\right)\cdot 211^{4} + \left(59 a^{2} + 67 a + 79\right)\cdot 211^{5} + \left(15 a^{2} + 10 a + 20\right)\cdot 211^{6} + \left(50 a^{2} + 114 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 |
| $(4,7)(5,6)(8,9)$ |
| $(1,9,8)(2,6,5)(3,4,7)$ |
| $(1,3)(4,6)(5,7)$ |
| $(4,6,9)(5,7,8)$ |
| $(1,3,2)(4,6,9)(5,8,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$6$ |
| $9$ |
$2$ |
$(1,3)(6,9)(7,8)$ |
$0$ |
| $9$ |
$2$ |
$(4,7)(5,6)(8,9)$ |
$2$ |
| $9$ |
$2$ |
$(1,6)(2,9)(3,4)(5,7)$ |
$0$ |
| $2$ |
$3$ |
$(1,3,2)(4,6,9)(5,8,7)$ |
$-3$ |
| $6$ |
$3$ |
$(1,8,6)(2,5,4)(3,7,9)$ |
$0$ |
| $6$ |
$3$ |
$(1,3,2)(4,9,6)$ |
$0$ |
| $12$ |
$3$ |
$(1,9,8)(2,6,5)(3,4,7)$ |
$0$ |
| $18$ |
$6$ |
$(1,9,8,3,6,7)(2,4,5)$ |
$0$ |
| $18$ |
$6$ |
$(1,3,2)(4,5,9,7,6,8)$ |
$-1$ |
| $18$ |
$6$ |
$(1,9,3,6,2,4)(5,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
