Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 191 }$ to precision 10.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 191 }$: $ x^{3} + 4 x + 172 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 87\cdot 191 + 153\cdot 191^{2} + 150\cdot 191^{3} + 9\cdot 191^{4} + 95\cdot 191^{5} + 9\cdot 191^{6} + 61\cdot 191^{7} + 62\cdot 191^{8} + 8\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 95 + 173\cdot 191 + 118\cdot 191^{2} + 113\cdot 191^{3} + 181\cdot 191^{4} + 25\cdot 191^{5} + 73\cdot 191^{6} + 52\cdot 191^{7} + 186\cdot 191^{8} + 29\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 112 + 83\cdot 191 + 104\cdot 191^{2} + 62\cdot 191^{3} + 138\cdot 191^{4} + 176\cdot 191^{5} + 70\cdot 191^{6} + 88\cdot 191^{7} + 33\cdot 191^{8} + 64\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 57 a^{2} + 87 a + 91 + \left(157 a^{2} + 135 a + 41\right)\cdot 191 + \left(146 a^{2} + 13 a + 126\right)\cdot 191^{2} + \left(118 a^{2} + 26 a + 138\right)\cdot 191^{3} + \left(133 a^{2} + 81 a + 153\right)\cdot 191^{4} + \left(175 a^{2} + 6 a + 65\right)\cdot 191^{5} + \left(53 a^{2} + 184 a + 143\right)\cdot 191^{6} + \left(81 a^{2} + 98 a + 153\right)\cdot 191^{7} + \left(170 a^{2} + 61 a + 57\right)\cdot 191^{8} + \left(164 a^{2} + 93 a + 133\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 96 a^{2} + 93 a + 114 + \left(113 a^{2} + 17 a + 56\right)\cdot 191 + \left(89 a^{2} + 125 a + 60\right)\cdot 191^{2} + \left(117 a^{2} + 73 a\right)\cdot 191^{3} + \left(133 a^{2} + 76 a + 67\right)\cdot 191^{4} + \left(113 a^{2} + 179 a + 97\right)\cdot 191^{5} + \left(186 a^{2} + 117 a + 128\right)\cdot 191^{6} + \left(36 a^{2} + 161 a + 30\right)\cdot 191^{7} + \left(163 a + 112\right)\cdot 191^{8} + \left(94 a^{2} + 30 a + 13\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 a^{2} + 135 a + \left(84 a^{2} + 26 a + 43\right)\cdot 191 + \left(70 a^{2} + 184 a + 9\right)\cdot 191^{2} + \left(190 a^{2} + 124 a + 131\right)\cdot 191^{3} + \left(80 a^{2} + 75 a + 117\right)\cdot 191^{4} + \left(117 a^{2} + 141 a + 43\right)\cdot 191^{5} + \left(32 a^{2} + 75 a + 36\right)\cdot 191^{6} + \left(a^{2} + 53 a + 126\right)\cdot 191^{7} + \left(77 a^{2} + 164 a + 189\right)\cdot 191^{8} + \left(188 a^{2} + 23 a + 137\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 a^{2} + 92 a + 42 + \left(146 a^{2} + 16 a + 77\right)\cdot 191 + \left(75 a^{2} + 122 a\right)\cdot 191^{2} + \left(72 a^{2} + 136 a + 15\right)\cdot 191^{3} + \left(98 a^{2} + 8 a + 187\right)\cdot 191^{4} + \left(88 a^{2} + 11 a + 87\right)\cdot 191^{5} + \left(144 a^{2} + 16 a + 66\right)\cdot 191^{6} + \left(123 a^{2} + 78 a + 12\right)\cdot 191^{7} + \left(140 a^{2} + 65 a + 42\right)\cdot 191^{8} + \left(34 a^{2} + 181 a + 168\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 167 a^{2} + 12 a + 66 + \left(77 a^{2} + 39 a + 84\right)\cdot 191 + \left(159 a^{2} + 55 a + 32\right)\cdot 191^{2} + \left(190 a^{2} + 28 a + 76\right)\cdot 191^{3} + \left(149 a^{2} + 101 a + 6\right)\cdot 191^{4} + \left(117 a^{2} + 173 a + 166\right)\cdot 191^{5} + \left(183 a^{2} + 181 a + 170\right)\cdot 191^{6} + \left(176 a^{2} + 13 a + 26\right)\cdot 191^{7} + \left(70 a^{2} + 64 a + 47\right)\cdot 191^{8} + \left(182 a^{2} + 107 a + 116\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
| $r_{ 9 }$ |
$=$ |
$ 185 a^{2} + 154 a + 33 + \left(183 a^{2} + 146 a + 117\right)\cdot 191 + \left(30 a^{2} + 72 a + 158\right)\cdot 191^{2} + \left(74 a^{2} + 183 a + 75\right)\cdot 191^{3} + \left(167 a^{2} + 38 a + 93\right)\cdot 191^{4} + \left(150 a^{2} + 61 a + 5\right)\cdot 191^{5} + \left(162 a^{2} + 188 a + 65\right)\cdot 191^{6} + \left(152 a^{2} + 166 a + 21\right)\cdot 191^{7} + \left(113 a^{2} + 53 a + 33\right)\cdot 191^{8} + \left(99 a^{2} + 136 a + 92\right)\cdot 191^{9} +O\left(191^{ 10 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 9 }$
| Cycle notation |
| $(1,3,2)(5,9,6)$ |
| $(1,5,7)(2,9,8)(3,6,4)$ |
| $(1,2)(4,7)(5,6)$ |
| $(4,6)(5,7)(8,9)$ |
| $(4,8,7)(5,9,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 9 }$
| Character value |
| $1$ | $1$ | $()$ | $6$ |
| $9$ | $2$ | $(1,2)(6,9)(7,8)$ | $0$ |
| $9$ | $2$ | $(4,6)(5,7)(8,9)$ | $-2$ |
| $9$ | $2$ | $(1,5)(2,6)(3,9)(4,7)$ | $0$ |
| $2$ | $3$ | $(1,2,3)(4,7,8)(5,9,6)$ | $-3$ |
| $6$ | $3$ | $(1,3,2)(5,9,6)$ | $0$ |
| $6$ | $3$ | $(1,7,9)(2,8,6)(3,4,5)$ | $0$ |
| $12$ | $3$ | $(1,5,7)(2,9,8)(3,6,4)$ | $0$ |
| $18$ | $6$ | $(1,6,7,2,9,8)(3,5,4)$ | $0$ |
| $18$ | $6$ | $(1,3,2)(4,5,7,9,8,6)$ | $1$ |
| $18$ | $6$ | $(1,6,3,5,2,9)(4,7)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
