Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 28 + 4\cdot 181 + 107\cdot 181^{2} + 93\cdot 181^{3} + 21\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 39 + 149\cdot 181 + 110\cdot 181^{2} + 16\cdot 181^{3} + 154\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 138\cdot 181 + 62\cdot 181^{2} + 60\cdot 181^{3} + 85\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 66 + 102\cdot 181 + 66\cdot 181^{2} + 164\cdot 181^{3} + 36\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 115 + 78\cdot 181 + 114\cdot 181^{2} + 16\cdot 181^{3} + 144\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 126 + 42\cdot 181 + 118\cdot 181^{2} + 120\cdot 181^{3} + 95\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 142 + 31\cdot 181 + 70\cdot 181^{2} + 164\cdot 181^{3} + 26\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 153 + 176\cdot 181 + 73\cdot 181^{2} + 87\cdot 181^{3} + 159\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,7)(4,5)(6,8)$ |
| $(1,2)(3,4)(5,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,7)(4,5)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,7,6,4)(2,3,5,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
