Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 107\cdot 181^{2} + 8\cdot 181^{3} + 117\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 156\cdot 181 + 155\cdot 181^{2} + 117\cdot 181^{3} + 118\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 155\cdot 181 + 139\cdot 181^{2} + 51\cdot 181^{3} + 77\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 81 + 145\cdot 181 + 141\cdot 181^{2} + 116\cdot 181^{3} + 8\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 114 + 102\cdot 181 + 107\cdot 181^{2} + 47\cdot 181^{3} + 23\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 134 + 138\cdot 181 + 137\cdot 181^{2} + 79\cdot 181^{3} + 30\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 136 + 113\cdot 181 + 5\cdot 181^{2} + 8\cdot 181^{3} + 32\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 140 + 92\cdot 181 + 109\cdot 181^{2} + 112\cdot 181^{3} + 135\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,7,8)(2,5,6,4)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,6)(3,8)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,7,8)(2,5,6,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
