Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 28\cdot 181 + 112\cdot 181^{2} + 63\cdot 181^{3} + 40\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 25 + 152\cdot 181 + 49\cdot 181^{2} + 76\cdot 181^{3} + 43\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 116\cdot 181 + 147\cdot 181^{2} + 89\cdot 181^{3} + 47\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 42 + 16\cdot 181 + 143\cdot 181^{2} + 166\cdot 181^{3} + 98\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 139 + 164\cdot 181 + 37\cdot 181^{2} + 14\cdot 181^{3} + 82\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 155 + 64\cdot 181 + 33\cdot 181^{2} + 91\cdot 181^{3} + 133\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 156 + 28\cdot 181 + 131\cdot 181^{2} + 104\cdot 181^{3} + 137\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 174 + 152\cdot 181 + 68\cdot 181^{2} + 117\cdot 181^{3} + 140\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,8,7)(3,5,6,4)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
