Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 + 61\cdot 181 + 137\cdot 181^{2} + 160\cdot 181^{3} + 54\cdot 181^{4} + 60\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 43 + 168\cdot 181 + 147\cdot 181^{2} + 19\cdot 181^{3} + 23\cdot 181^{4} + 176\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 56 + 19\cdot 181 + 70\cdot 181^{2} + 141\cdot 181^{3} + 48\cdot 181^{4} + 177\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 59 + 139\cdot 181 + 141\cdot 181^{2} + 141\cdot 181^{3} + 153\cdot 181^{4} + 71\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 81 + 141\cdot 181 + 159\cdot 181^{2} + 88\cdot 181^{3} + 72\cdot 181^{4} + 88\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 122 + 48\cdot 181 + 115\cdot 181^{2} + 71\cdot 181^{3} + 157\cdot 181^{4} + 112\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 158 + 51\cdot 181 + 39\cdot 181^{2} + 169\cdot 181^{3} + 100\cdot 181^{4} + 11\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 180 + 93\cdot 181 + 93\cdot 181^{2} + 111\cdot 181^{3} + 112\cdot 181^{4} + 25\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(3,6)(5,8)$ |
| $(1,8,6,2,7,5,3,4)$ |
| $(2,4)(3,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,7)(2,4)(3,6)(5,8)$ | $-4$ |
| $2$ | $2$ | $(2,4)(5,8)$ | $0$ |
| $4$ | $2$ | $(2,4)(3,6)$ | $0$ |
| $4$ | $2$ | $(1,6)(2,8)(3,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,6,7,3)(2,5,4,8)$ | $0$ |
| $2$ | $4$ | $(1,6,7,3)(2,8,4,5)$ | $0$ |
| $4$ | $8$ | $(1,8,6,2,7,5,3,4)$ | $0$ |
| $4$ | $8$ | $(1,2,3,8,7,4,6,5)$ | $0$ |
| $4$ | $8$ | $(1,8,6,4,7,5,3,2)$ | $0$ |
| $4$ | $8$ | $(1,4,3,8,7,2,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
