Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 94\cdot 181 + 33\cdot 181^{2} + 108\cdot 181^{3} + 46\cdot 181^{4} +O\left(181^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 16 + 176\cdot 181 + 13\cdot 181^{2} + 64\cdot 181^{3} + 31\cdot 181^{4} + 108\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 30 + 91\cdot 181 + 94\cdot 181^{2} + 99\cdot 181^{3} + 28\cdot 181^{4} + 132\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 48\cdot 181 + 49\cdot 181^{2} + 77\cdot 181^{3} + 16\cdot 181^{4} + 73\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 82 + 42\cdot 181 + 97\cdot 181^{2} + 129\cdot 181^{4} + 81\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 105 + 159\cdot 181 + 18\cdot 181^{2} + 89\cdot 181^{3} + 79\cdot 181^{4} + 120\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 108 + 114\cdot 181 + 151\cdot 181^{2} + 106\cdot 181^{3} + 42\cdot 181^{4} + 101\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 155 + 178\cdot 181 + 83\cdot 181^{2} + 178\cdot 181^{3} + 168\cdot 181^{4} + 106\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4,2,5,7,6,3,8)$ |
| $(2,3)(5,8)$ |
| $(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,3)(4,6)(5,8)$ | $-4$ |
| $2$ | $2$ | $(4,6)(5,8)$ | $0$ |
| $4$ | $2$ | $(2,3)(5,8)$ | $0$ |
| $4$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,7,3)(4,5,6,8)$ | $0$ |
| $2$ | $4$ | $(1,2,7,3)(4,8,6,5)$ | $0$ |
| $4$ | $8$ | $(1,4,2,5,7,6,3,8)$ | $0$ |
| $4$ | $8$ | $(1,5,3,4,7,8,2,6)$ | $0$ |
| $4$ | $8$ | $(1,4,2,8,7,6,3,5)$ | $0$ |
| $4$ | $8$ | $(1,8,3,4,7,5,2,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
