Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 211 }$ to precision 14.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 211 }$: $ x^{3} + 2 x + 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} + 29\cdot 211^{10} + 192\cdot 211^{11} + 100\cdot 211^{12} + 140\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 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} + 86\cdot 211^{10} + 191\cdot 211^{11} + 142\cdot 211^{12} + 86\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 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} + 94\cdot 211^{10} + 38\cdot 211^{11} + 178\cdot 211^{12} + 194\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 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} + \left(9 a^{2} + 111 a + 83\right)\cdot 211^{10} + \left(16 a^{2} + 139 a + 21\right)\cdot 211^{11} + \left(159 a^{2} + 46 a + 1\right)\cdot 211^{12} + \left(200 a^{2} + 164 a + 127\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 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} + \left(145 a^{2} + 148 a + 193\right)\cdot 211^{10} + \left(47 a^{2} + 210 a + 133\right)\cdot 211^{11} + \left(77 a^{2} + 99 a + 32\right)\cdot 211^{12} + \left(52 a^{2} + 146 a + 140\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 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} + \left(95 a^{2} + 176 a + 197\right)\cdot 211^{10} + \left(15 a^{2} + 152 a + 90\right)\cdot 211^{11} + \left(86 a^{2} + 52 a + 44\right)\cdot 211^{12} + \left(173 a^{2} + 10 a + 20\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 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} + \left(56 a^{2} + 161 a + 145\right)\cdot 211^{10} + \left(147 a^{2} + 71 a + 55\right)\cdot 211^{11} + \left(185 a^{2} + 64 a + 177\right)\cdot 211^{12} + \left(168 a^{2} + 111 a + 154\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 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} + \left(12 a^{2} + 195 a + 17\right)\cdot 211^{10} + \left(91 a^{2} + 74 a + 51\right)\cdot 211^{11} + \left(79 a^{2} + 69 a + 176\right)\cdot 211^{12} + \left(209 a^{2} + 195 a + 208\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 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} + \left(103 a^{2} + 49 a + 207\right)\cdot 211^{10} + \left(104 a^{2} + 194 a + 68\right)\cdot 211^{11} + \left(45 a^{2} + 88 a + 201\right)\cdot 211^{12} + \left(39 a^{2} + 5 a + 192\right)\cdot 211^{13} +O\left(211^{ 14 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(6,8,9)$ |
| $(1,2,3)$ |
| $(4,9)(5,6)(7,8)$ |
| $(4,5,7)$ |
| $(1,6)(2,8)(3,9)(4,7,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $9$ | $2$ | $(1,6)(2,8)(3,9)$ | $-1$ |
| $1$ | $3$ | $(1,2,3)(4,5,7)(6,8,9)$ | $3 \zeta_{3}$ |
| $1$ | $3$ | $(1,3,2)(4,7,5)(6,9,8)$ | $-3 \zeta_{3} - 3$ |
| $3$ | $3$ | $(1,2,3)$ | $\zeta_{3} - 1$ |
| $3$ | $3$ | $(1,3,2)$ | $-\zeta_{3} - 2$ |
| $3$ | $3$ | $(1,3,2)(4,5,7)(6,9,8)$ | $2 \zeta_{3} + 1$ |
| $3$ | $3$ | $(1,2,3)(4,7,5)(6,8,9)$ | $-2 \zeta_{3} - 1$ |
| $3$ | $3$ | $(1,2,3)(4,5,7)$ | $-\zeta_{3} + 1$ |
| $3$ | $3$ | $(1,3,2)(4,7,5)$ | $\zeta_{3} + 2$ |
| $6$ | $3$ | $(1,3,2)(4,5,7)$ | $0$ |
| $18$ | $3$ | $(1,7,6)(2,4,8)(3,5,9)$ | $0$ |
| $9$ | $6$ | $(1,6)(2,8)(3,9)(4,7,5)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,6)(2,8)(3,9)(4,5,7)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,8,2,9,3,6)(4,7,5)$ | $\zeta_{3} + 1$ |
| $9$ | $6$ | $(1,6,3,9,2,8)(4,5,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,9,3,8,2,6)(4,7,5)$ | $-1$ |
| $9$ | $6$ | $(1,6,2,8,3,9)(4,5,7)$ | $-1$ |
| $9$ | $6$ | $(1,4,2,5,3,7)$ | $-\zeta_{3}$ |
| $9$ | $6$ | $(1,7,3,5,2,4)$ | $\zeta_{3} + 1$ |
| $18$ | $9$ | $(1,5,9,3,4,8,2,7,6)$ | $0$ |
| $18$ | $9$ | $(1,9,4,2,6,5,3,8,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
