Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 17.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 a + 151 + \left(158 a + 132\right)\cdot 167 + \left(105 a + 38\right)\cdot 167^{2} + \left(19 a + 8\right)\cdot 167^{3} + \left(154 a + 117\right)\cdot 167^{4} + \left(133 a + 45\right)\cdot 167^{5} + \left(164 a + 38\right)\cdot 167^{6} + \left(99 a + 6\right)\cdot 167^{7} + \left(36 a + 144\right)\cdot 167^{8} + \left(8 a + 74\right)\cdot 167^{9} + \left(22 a + 140\right)\cdot 167^{10} + \left(18 a + 34\right)\cdot 167^{11} + \left(20 a + 113\right)\cdot 167^{12} + \left(116 a + 22\right)\cdot 167^{13} + 22\cdot 167^{14} + \left(120 a + 89\right)\cdot 167^{15} + \left(160 a + 138\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 105 + 49\cdot 167 + 140\cdot 167^{2} + 35\cdot 167^{3} + 154\cdot 167^{4} + 110\cdot 167^{5} + 97\cdot 167^{6} + 160\cdot 167^{7} + 15\cdot 167^{8} + 89\cdot 167^{9} + 42\cdot 167^{10} + 153\cdot 167^{11} + 139\cdot 167^{12} + 113\cdot 167^{13} + 15\cdot 167^{14} + 47\cdot 167^{15} + 104\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 a + 25 + \left(102 a + 125\right)\cdot 167 + \left(33 a + 7\right)\cdot 167^{2} + \left(37 a + 147\right)\cdot 167^{3} + \left(56 a + 43\right)\cdot 167^{4} + \left(78 a + 104\right)\cdot 167^{5} + \left(145 a + 96\right)\cdot 167^{6} + \left(72 a + 100\right)\cdot 167^{7} + \left(97 a + 151\right)\cdot 167^{8} + \left(165 a + 143\right)\cdot 167^{9} + \left(44 a + 123\right)\cdot 167^{10} + \left(144 a + 117\right)\cdot 167^{11} + \left(150 a + 92\right)\cdot 167^{12} + \left(147 a + 21\right)\cdot 167^{13} + \left(36 a + 97\right)\cdot 167^{14} + \left(35 a + 43\right)\cdot 167^{15} + \left(129 a + 80\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ a + 53 + \left(40 a + 106\right)\cdot 167 + \left(135 a + 89\right)\cdot 167^{2} + \left(154 a + 9\right)\cdot 167^{3} + \left(84 a + 61\right)\cdot 167^{4} + \left(147 a + 147\right)\cdot 167^{5} + \left(153 a + 137\right)\cdot 167^{6} + \left(93 a + 143\right)\cdot 167^{7} + \left(155 a + 9\right)\cdot 167^{8} + \left(81 a + 3\right)\cdot 167^{9} + \left(14 a + 39\right)\cdot 167^{10} + \left(130 a + 162\right)\cdot 167^{11} + \left(20 a + 151\right)\cdot 167^{12} + \left(132 a + 158\right)\cdot 167^{13} + \left(100 a + 84\right)\cdot 167^{14} + \left(84 a + 133\right)\cdot 167^{15} + \left(37 a + 140\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 161 a + 157 + \left(8 a + 117\right)\cdot 167 + \left(61 a + 153\right)\cdot 167^{2} + \left(147 a + 88\right)\cdot 167^{3} + \left(12 a + 84\right)\cdot 167^{4} + \left(33 a + 25\right)\cdot 167^{5} + \left(2 a + 69\right)\cdot 167^{6} + \left(67 a + 108\right)\cdot 167^{7} + \left(130 a + 80\right)\cdot 167^{8} + \left(158 a + 46\right)\cdot 167^{9} + \left(144 a + 154\right)\cdot 167^{10} + \left(148 a + 30\right)\cdot 167^{11} + \left(146 a + 115\right)\cdot 167^{12} + \left(50 a + 118\right)\cdot 167^{13} + \left(166 a + 73\right)\cdot 167^{14} + \left(46 a + 41\right)\cdot 167^{15} + \left(6 a + 12\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 104 a + 88 + \left(64 a + 164\right)\cdot 167 + \left(133 a + 105\right)\cdot 167^{2} + \left(129 a + 150\right)\cdot 167^{3} + \left(110 a + 62\right)\cdot 167^{4} + \left(88 a + 126\right)\cdot 167^{5} + \left(21 a + 163\right)\cdot 167^{6} + \left(94 a + 27\right)\cdot 167^{7} + \left(69 a + 9\right)\cdot 167^{8} + \left(a + 45\right)\cdot 167^{9} + \left(122 a + 3\right)\cdot 167^{10} + \left(22 a + 50\right)\cdot 167^{11} + \left(16 a + 99\right)\cdot 167^{12} + \left(19 a + 18\right)\cdot 167^{13} + \left(130 a + 153\right)\cdot 167^{14} + \left(131 a + 41\right)\cdot 167^{15} + \left(37 a + 7\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 + 160\cdot 167 + 113\cdot 167^{2} + 31\cdot 167^{3} + 153\cdot 167^{4} + 64\cdot 167^{5} + 87\cdot 167^{6} + 36\cdot 167^{7} + 18\cdot 167^{8} + 2\cdot 167^{9} + 26\cdot 167^{10} + 8\cdot 167^{11} + 80\cdot 167^{12} + 110\cdot 167^{13} + 154\cdot 167^{15} + 90\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 166 a + 54 + \left(126 a + 145\right)\cdot 167 + \left(31 a + 17\right)\cdot 167^{2} + \left(12 a + 29\right)\cdot 167^{3} + \left(82 a + 158\right)\cdot 167^{4} + \left(19 a + 42\right)\cdot 167^{5} + \left(13 a + 144\right)\cdot 167^{6} + \left(73 a + 83\right)\cdot 167^{7} + \left(11 a + 71\right)\cdot 167^{8} + \left(85 a + 96\right)\cdot 167^{9} + \left(152 a + 138\right)\cdot 167^{10} + \left(36 a + 110\right)\cdot 167^{11} + \left(146 a + 42\right)\cdot 167^{12} + \left(34 a + 103\right)\cdot 167^{13} + \left(66 a + 53\right)\cdot 167^{14} + \left(82 a + 117\right)\cdot 167^{15} + \left(129 a + 93\right)\cdot 167^{16} +O\left(167^{ 17 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,4)(5,8,7,6)$ |
| $(1,2)(7,8)$ |
| $(1,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-4$ |
| $4$ | $2$ | $(3,6)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,6)(4,5)$ | $2$ |
| $6$ | $2$ | $(1,8)(3,6)$ | $0$ |
| $12$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $12$ | $2$ | $(1,2)(7,8)$ | $-2$ |
| $12$ | $2$ | $(1,4)(2,7)(3,6)(5,8)$ | $2$ |
| $24$ | $2$ | $(1,2)(3,6)(7,8)$ | $0$ |
| $32$ | $3$ | $(1,3,4)(5,8,6)$ | $1$ |
| $12$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $12$ | $4$ | $(1,2,8,7)$ | $2$ |
| $12$ | $4$ | $(1,8)(2,7)(3,5,6,4)$ | $-2$ |
| $24$ | $4$ | $(1,3,8,6)(2,4)(5,7)$ | $0$ |
| $24$ | $4$ | $(1,2,8,7)(3,6)$ | $0$ |
| $48$ | $4$ | $(1,2,3,4)(5,8,7,6)$ | $0$ |
| $32$ | $6$ | $(2,4,3,7,5,6)$ | $-1$ |
| $32$ | $6$ | $(1,3,4)(2,7)(5,8,6)$ | $1$ |
| $32$ | $6$ | $(1,3,5,8,6,4)(2,7)$ | $-1$ |
| $48$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
