Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 181 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 22 + 175\cdot 181 + 79\cdot 181^{2} + 7\cdot 181^{3} + 73\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 75 + 83\cdot 181 + 155\cdot 181^{2} + 167\cdot 181^{3} + 175\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 78 + 20\cdot 181 + 15\cdot 181^{2} + 167\cdot 181^{3} + 131\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 79 + 152\cdot 181 + 68\cdot 181^{2} + 50\cdot 181^{3} + 24\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 93 + 92\cdot 181 + 125\cdot 181^{2} + 34\cdot 181^{3} + 158\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 95 + 110\cdot 181 + 92\cdot 181^{2} + 35\cdot 181^{3} + 103\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 119 + 106\cdot 181 + 23\cdot 181^{2} + 91\cdot 181^{3} + 51\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 166 + 163\cdot 181 + 162\cdot 181^{2} + 169\cdot 181^{3} + 5\cdot 181^{4} +O\left(181^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,6,7,4)(2,8,5,3)$ |
| $(1,8,4,2,7,3,6,5)$ |
| $(1,2)(3,6)(4,8)(5,7)$ |
| $(1,7)(2,5)(3,8)(4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,5)(3,8)(4,6)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(1,4)(2,5)(6,7)$ |
$0$ |
$0$ |
| $4$ |
$2$ |
$(1,2)(3,6)(4,8)(5,7)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,5,8)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,8,4,2,7,3,6,5)$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ |
$8$ |
$(1,2,6,8,7,5,4,3)$ |
$\zeta_{8}^{3} - \zeta_{8}$ |
$-\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
