Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 167 }$ to precision 13.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 167 }$: $ x^{2} + 166 x + 5 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 96 a + 131 + \left(51 a + 142\right)\cdot 167 + \left(13 a + 65\right)\cdot 167^{2} + \left(142 a + 28\right)\cdot 167^{3} + \left(147 a + 32\right)\cdot 167^{4} + \left(33 a + 38\right)\cdot 167^{5} + \left(33 a + 152\right)\cdot 167^{6} + \left(109 a + 125\right)\cdot 167^{7} + \left(144 a + 69\right)\cdot 167^{8} + \left(49 a + 28\right)\cdot 167^{9} + \left(37 a + 138\right)\cdot 167^{10} + \left(74 a + 51\right)\cdot 167^{11} + \left(114 a + 94\right)\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 71 a + 60 + \left(115 a + 98\right)\cdot 167 + \left(153 a + 27\right)\cdot 167^{2} + \left(24 a + 157\right)\cdot 167^{3} + \left(19 a + 37\right)\cdot 167^{4} + \left(133 a + 91\right)\cdot 167^{5} + \left(133 a + 151\right)\cdot 167^{6} + \left(57 a + 34\right)\cdot 167^{7} + \left(22 a + 105\right)\cdot 167^{8} + \left(117 a + 100\right)\cdot 167^{9} + \left(129 a + 125\right)\cdot 167^{10} + \left(92 a + 88\right)\cdot 167^{11} + \left(52 a + 134\right)\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 45 a + 23 + \left(31 a + 44\right)\cdot 167 + \left(53 a + 119\right)\cdot 167^{2} + \left(143 a + 53\right)\cdot 167^{3} + \left(161 a + 66\right)\cdot 167^{4} + \left(137 a + 22\right)\cdot 167^{5} + 24\cdot 167^{6} + \left(121 a + 95\right)\cdot 167^{7} + \left(73 a + 28\right)\cdot 167^{8} + \left(80 a + 69\right)\cdot 167^{9} + \left(47 a + 18\right)\cdot 167^{10} + \left(139 a + 68\right)\cdot 167^{11} + \left(62 a + 49\right)\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 56 a + 91 + \left(91 a + 77\right)\cdot 167 + \left(135 a + 165\right)\cdot 167^{2} + \left(82 a + 71\right)\cdot 167^{3} + \left(117 a + 61\right)\cdot 167^{4} + \left(134 a + 112\right)\cdot 167^{5} + \left(27 a + 156\right)\cdot 167^{6} + \left(99 a + 127\right)\cdot 167^{7} + \left(68 a + 148\right)\cdot 167^{8} + \left(162 a + 37\right)\cdot 167^{9} + \left(13 a + 60\right)\cdot 167^{10} + \left(65 a + 85\right)\cdot 167^{11} + 18\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 7 + 15\cdot 167 + 165\cdot 167^{2} + 107\cdot 167^{3} + 51\cdot 167^{4} + 51\cdot 167^{5} + 96\cdot 167^{6} + 164\cdot 167^{7} + 155\cdot 167^{8} + 22\cdot 167^{9} + 81\cdot 167^{10} + 32\cdot 167^{11} + 124\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 143 + 146\cdot 167 + 107\cdot 167^{2} + 85\cdot 167^{3} + 70\cdot 167^{4} + 57\cdot 167^{5} + 150\cdot 167^{6} + 38\cdot 167^{7} + 60\cdot 167^{8} + 34\cdot 167^{9} + 13\cdot 167^{10} + 45\cdot 167^{11} + 153\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 111 a + 147 + \left(75 a + 112\right)\cdot 167 + \left(31 a + 42\right)\cdot 167^{2} + \left(84 a + 19\right)\cdot 167^{3} + \left(49 a + 96\right)\cdot 167^{4} + \left(32 a + 129\right)\cdot 167^{5} + \left(139 a + 49\right)\cdot 167^{6} + \left(67 a + 32\right)\cdot 167^{7} + \left(98 a + 118\right)\cdot 167^{8} + \left(4 a + 131\right)\cdot 167^{9} + \left(153 a + 78\right)\cdot 167^{10} + \left(101 a + 136\right)\cdot 167^{11} + \left(166 a + 120\right)\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 122 a + 68 + \left(135 a + 30\right)\cdot 167 + \left(113 a + 141\right)\cdot 167^{2} + \left(23 a + 143\right)\cdot 167^{3} + \left(5 a + 84\right)\cdot 167^{4} + \left(29 a + 165\right)\cdot 167^{5} + \left(166 a + 53\right)\cdot 167^{6} + \left(45 a + 48\right)\cdot 167^{7} + \left(93 a + 148\right)\cdot 167^{8} + \left(86 a + 75\right)\cdot 167^{9} + \left(119 a + 152\right)\cdot 167^{10} + \left(27 a + 159\right)\cdot 167^{11} + \left(104 a + 139\right)\cdot 167^{12} +O\left(167^{ 13 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5)(2,6)(3,4)(7,8)$ |
| $(3,8)$ |
| $(4,7)$ |
| $(1,2)$ |
| $(5,6)$ |
| $(1,7,8,2,4,3)(5,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,2)(3,8)(4,7)(5,6)$ |
$-4$ |
| $4$ |
$2$ |
$(4,7)$ |
$2$ |
| $4$ |
$2$ |
$(1,2)(3,8)(5,6)$ |
$-2$ |
| $6$ |
$2$ |
$(3,8)(4,7)$ |
$0$ |
| $12$ |
$2$ |
$(1,5)(2,6)(3,4)(7,8)$ |
$0$ |
| $16$ |
$3$ |
$(1,4,8)(2,7,3)$ |
$1$ |
| $16$ |
$3$ |
$(1,8,4)(2,3,7)$ |
$1$ |
| $12$ |
$4$ |
$(1,6,2,5)(3,4,8,7)$ |
$0$ |
| $24$ |
$4$ |
$(1,5)(2,6)(3,4,8,7)$ |
$0$ |
| $16$ |
$6$ |
$(1,3,4,2,8,7)(5,6)$ |
$-1$ |
| $16$ |
$6$ |
$(1,7,8,2,4,3)(5,6)$ |
$-1$ |
| $16$ |
$6$ |
$(1,3,7)(2,8,4)(5,6)$ |
$-1$ |
| $16$ |
$6$ |
$(1,7,3)(2,4,8)(5,6)$ |
$-1$ |
| $16$ |
$6$ |
$(1,8,5,2,3,6)$ |
$1$ |
| $16$ |
$6$ |
$(1,6,3,2,5,8)$ |
$1$ |
The blue line marks the conjugacy class containing complex conjugation.
