Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 157\cdot 181 + 151\cdot 181^{2} + 12\cdot 181^{3} + 166\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 62 + 69\cdot 181 + 107\cdot 181^{2} + 135\cdot 181^{3} + 44\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 80 + 127\cdot 181 + 169\cdot 181^{2} + 101\cdot 181^{3} + 11\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 88 + 59\cdot 181 + 124\cdot 181^{2} + 45\cdot 181^{3} + 47\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 + 121\cdot 181 + 56\cdot 181^{2} + 135\cdot 181^{3} + 133\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 101 + 53\cdot 181 + 11\cdot 181^{2} + 79\cdot 181^{3} + 169\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 119 + 111\cdot 181 + 73\cdot 181^{2} + 45\cdot 181^{3} + 136\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 149 + 23\cdot 181 + 29\cdot 181^{2} + 168\cdot 181^{3} + 14\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
