Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 162\cdot 163 + 150\cdot 163^{2} + 20\cdot 163^{3} + 10\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 134\cdot 163 + 120\cdot 163^{2} + 24\cdot 163^{3} + 62\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 39 + 85\cdot 163 + 113\cdot 163^{2} + 108\cdot 163^{3} + 147\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 52 + 72\cdot 163 + 48\cdot 163^{2} + 155\cdot 163^{3} + 119\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 58 + 115\cdot 163 + 21\cdot 163^{2} + 157\cdot 163^{3} + 31\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 72 + 108\cdot 163 + 23\cdot 163^{2} + 104\cdot 163^{3} + 44\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 109 + 107\cdot 163 + 103\cdot 163^{2} + 8\cdot 163^{3} + 106\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 145 + 29\cdot 163 + 69\cdot 163^{2} + 72\cdot 163^{3} + 129\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,2,7)(4,6,5,8)$ |
| $(1,5,2,4)(3,6,7,8)$ |
| $(1,2)(3,7)(4,5)(6,8)$ |
| $(1,3)(2,7)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,5)(6,8)$ | $-2$ |
| $4$ | $2$ | $(1,3)(2,7)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,3,2,7)(4,6,5,8)$ | $0$ |
| $4$ | $4$ | $(1,5,2,4)(3,6,7,8)$ | $0$ |
| $2$ | $8$ | $(1,4,3,6,2,5,7,8)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,8,2,4,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
