Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 7.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{3} + 2 x + 209 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 13 + 89\cdot 211 + 207\cdot 211^{2} + 71\cdot 211^{3} + 191\cdot 211^{4} + 153\cdot 211^{5} + 194\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 156\cdot 211 + 51\cdot 211^{2} + 150\cdot 211^{3} + 79\cdot 211^{4} + 54\cdot 211^{5} + 46\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 182 + 176\cdot 211 + 162\cdot 211^{2} + 199\cdot 211^{3} + 150\cdot 211^{4} + 2\cdot 211^{5} + 181\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 39 a^{2} + 97 a + 52 + \left(111 a^{2} + 88 a + 148\right)\cdot 211 + \left(42 a^{2} + 85 a + 56\right)\cdot 211^{2} + \left(193 a^{2} + 60 a + 187\right)\cdot 211^{3} + \left(34 a^{2} + 183 a + 116\right)\cdot 211^{4} + \left(179 a^{2} + 87 a + 168\right)\cdot 211^{5} + \left(93 a^{2} + 56 a + 54\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 45 a^{2} + 88 a + 60 + \left(27 a^{2} + 100 a + 36\right)\cdot 211 + \left(48 a^{2} + 124 a + 64\right)\cdot 211^{2} + \left(11 a^{2} + 198 a + 85\right)\cdot 211^{3} + \left(26 a^{2} + 6 a + 175\right)\cdot 211^{4} + \left(21 a^{2} + 53 a + 168\right)\cdot 211^{5} + \left(35 a^{2} + 92 a + 46\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 48 a^{2} + 22 a + 64 + \left(65 a^{2} + 197 a + 157\right)\cdot 211 + \left(20 a^{2} + 90 a + 167\right)\cdot 211^{2} + \left(97 a^{2} + 161 a + 199\right)\cdot 211^{3} + \left(164 a^{2} + 62 a + 148\right)\cdot 211^{4} + \left(188 a^{2} + 91 a + 40\right)\cdot 211^{5} + \left(158 a^{2} + 22 a + 71\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 124 a^{2} + 92 a + 95 + \left(34 a^{2} + 136 a + 116\right)\cdot 211 + \left(148 a^{2} + 34 a + 197\right)\cdot 211^{2} + \left(131 a^{2} + 200 a + 34\right)\cdot 211^{3} + \left(11 a^{2} + 175 a + 156\right)\cdot 211^{4} + \left(54 a^{2} + 31 a + 1\right)\cdot 211^{5} + \left(169 a^{2} + 132 a + 85\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 169 a^{2} + 157 a + 155 + \left(104 a + 141\right)\cdot 211 + \left(57 a^{2} + 109 a + 5\right)\cdot 211^{2} + \left(167 a^{2} + 83 a + 12\right)\cdot 211^{3} + \left(54 a^{2} + 158 a + 73\right)\cdot 211^{4} + \left(62 a^{2} + 40 a + 153\right)\cdot 211^{5} + \left(105 a^{2} + 7 a + 210\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 208 a^{2} + 177 a + 207 + \left(182 a^{2} + 5 a + 32\right)\cdot 211 + \left(105 a^{2} + 188 a + 141\right)\cdot 211^{2} + \left(32 a^{2} + 139 a + 113\right)\cdot 211^{3} + \left(130 a^{2} + 45 a + 173\right)\cdot 211^{4} + \left(127 a^{2} + 117 a + 99\right)\cdot 211^{5} + \left(70 a^{2} + 111 a + 164\right)\cdot 211^{6} +O\left(211^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,2,3)(4,6,7)(5,8,9)$ |
| $(4,9)(5,6)(7,8)$ |
| $(2,3)(4,6)(5,9)$ |
| $(1,9,4)(2,5,6)(3,8,7)$ |
| $(4,6,7)(5,9,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(2,3)(6,7)(8,9)$ | $0$ |
| $9$ | $2$ | $(4,9)(5,6)(7,8)$ | $-2$ |
| $9$ | $2$ | $(1,8)(2,5)(3,9)(4,6)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,6,7)(5,8,9)$ | $-3$ |
| $6$ | $3$ | $(1,4,5)(2,6,8)(3,7,9)$ | $0$ |
| $6$ | $3$ | $(1,3,2)(5,8,9)$ | $0$ |
| $12$ | $3$ | $(1,9,4)(2,5,6)(3,8,7)$ | $0$ |
| $18$ | $6$ | $(1,5,4)(2,9,6,3,8,7)$ | $0$ |
| $18$ | $6$ | $(1,2,3)(4,5,7,9,6,8)$ | $1$ |
| $18$ | $6$ | $(1,5,3,8,2,9)(4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
