Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 53 + 138\cdot 163 + 59\cdot 163^{2} + 54\cdot 163^{3} + 73\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 59 + 36\cdot 163 + 66\cdot 163^{2} + 55\cdot 163^{3} + 156\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 60 + 75\cdot 163 + 45\cdot 163^{2} + 131\cdot 163^{3} + 3\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 154 + 75\cdot 163 + 154\cdot 163^{2} + 84\cdot 163^{3} + 92\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2,3,4)$ |
| $(1,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$3$ |
| $3$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
| $6$ |
$2$ |
$(1,2)$ |
$1$ |
| $8$ |
$3$ |
$(1,2,3)$ |
$0$ |
| $6$ |
$4$ |
$(1,2,3,4)$ |
$-1$ |
The blue line marks the conjugacy class containing complex conjugation.
