Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 169\cdot 181 + 133\cdot 181^{2} + 155\cdot 181^{3} + 138\cdot 181^{4} + 156\cdot 181^{5} + 47\cdot 181^{6} + 112\cdot 181^{7} + 88\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 48 + 94\cdot 181 + 107\cdot 181^{2} + 115\cdot 181^{3} + 51\cdot 181^{4} + 136\cdot 181^{5} + 88\cdot 181^{6} + 22\cdot 181^{7} + 133\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 53 + 111\cdot 181 + 104\cdot 181^{2} + 113\cdot 181^{3} + 21\cdot 181^{4} + 166\cdot 181^{5} + 170\cdot 181^{6} + 134\cdot 181^{7} + 100\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 136\cdot 181 + 88\cdot 181^{2} + 29\cdot 181^{3} + 165\cdot 181^{4} + 27\cdot 181^{5} + 154\cdot 181^{6} + 115\cdot 181^{7} + 92\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 102 + 44\cdot 181 + 92\cdot 181^{2} + 151\cdot 181^{3} + 15\cdot 181^{4} + 153\cdot 181^{5} + 26\cdot 181^{6} + 65\cdot 181^{7} + 88\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 128 + 69\cdot 181 + 76\cdot 181^{2} + 67\cdot 181^{3} + 159\cdot 181^{4} + 14\cdot 181^{5} + 10\cdot 181^{6} + 46\cdot 181^{7} + 80\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 133 + 86\cdot 181 + 73\cdot 181^{2} + 65\cdot 181^{3} + 129\cdot 181^{4} + 44\cdot 181^{5} + 92\cdot 181^{6} + 158\cdot 181^{7} + 47\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 174 + 11\cdot 181 + 47\cdot 181^{2} + 25\cdot 181^{3} + 42\cdot 181^{4} + 24\cdot 181^{5} + 133\cdot 181^{6} + 68\cdot 181^{7} + 92\cdot 181^{8} +O\left(181^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,7,5,8,3,2,4)$ |
| $(2,7)(4,5)$ |
| $(3,6)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,7)(2,8)(3,4)(5,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,5,8,3,2,4)$ |
$0$ |
| $4$ |
$8$ |
$(1,5,2,6,8,4,7,3)$ |
$0$ |
| $4$ |
$8$ |
$(1,6,7,4,8,3,2,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,4,2,6,8,5,7,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
