Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 16 + 104\cdot 163 + 155\cdot 163^{2} + 29\cdot 163^{3} + 5\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 34 + 151\cdot 163 + 60\cdot 163^{2} + 130\cdot 163^{3} + 82\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 35 + 79\cdot 163 + 156\cdot 163^{2} + 2\cdot 163^{3} + 122\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 74 + 7\cdot 163 + 120\cdot 163^{2} + 89\cdot 163^{3} + 97\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 89 + 155\cdot 163 + 42\cdot 163^{2} + 73\cdot 163^{3} + 65\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 128 + 83\cdot 163 + 6\cdot 163^{2} + 160\cdot 163^{3} + 40\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 129 + 11\cdot 163 + 102\cdot 163^{2} + 32\cdot 163^{3} + 80\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 147 + 58\cdot 163 + 7\cdot 163^{2} + 133\cdot 163^{3} + 157\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,8,6)(2,4,7,5)$ |
| $(1,2)(3,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
