Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 19 + 69\cdot 181 + 52\cdot 181^{2} + 46\cdot 181^{3} + 84\cdot 181^{4} + 33\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 23 + 48\cdot 181 + 36\cdot 181^{2} + 118\cdot 181^{3} + 25\cdot 181^{4} + 60\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 41 + 84\cdot 181 + 124\cdot 181^{2} + 100\cdot 181^{3} + 7\cdot 181^{4} + 46\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 102 + 81\cdot 181 + 139\cdot 181^{2} + 120\cdot 181^{3} + 69\cdot 181^{4} + 155\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 103 + 92\cdot 181 + 144\cdot 181^{2} + 64\cdot 181^{3} + 38\cdot 181^{4} + 113\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 122 + 72\cdot 181 + 154\cdot 181^{2} + 74\cdot 181^{3} + 15\cdot 181^{4} + 13\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 154 + 161\cdot 181 + 124\cdot 181^{2} + 153\cdot 181^{3} + 87\cdot 181^{4} + 50\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 163 + 113\cdot 181 + 128\cdot 181^{2} + 44\cdot 181^{3} + 33\cdot 181^{4} + 71\cdot 181^{5} +O\left(181^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(4,6)(7,8)$ |
| $(1,5,8,6,3,2,7,4)$ |
| $(2,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $4$ |
| $1$ | $2$ | $(1,3)(2,5)(4,6)(7,8)$ | $-4$ |
| $2$ | $2$ | $(2,5)(4,6)$ | $0$ |
| $4$ | $2$ | $(4,6)(7,8)$ | $0$ |
| $4$ | $2$ | $(1,8)(2,4)(3,7)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,3,7)(2,4,5,6)$ | $0$ |
| $2$ | $4$ | $(1,8,3,7)(2,6,5,4)$ | $0$ |
| $4$ | $8$ | $(1,5,8,6,3,2,7,4)$ | $0$ |
| $4$ | $8$ | $(1,6,7,5,3,4,8,2)$ | $0$ |
| $4$ | $8$ | $(1,5,8,4,3,2,7,6)$ | $0$ |
| $4$ | $8$ | $(1,4,7,5,3,6,8,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
