Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 163 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12 + 117\cdot 163 + 128\cdot 163^{2} + 77\cdot 163^{3} + 44\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 24 + 111\cdot 163 + 52\cdot 163^{2} + 21\cdot 163^{3} + 143\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 54 + 155\cdot 163 + 23\cdot 163^{2} + 125\cdot 163^{3} + 47\cdot 163^{4} +O\left(163^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 73 + 105\cdot 163 + 120\cdot 163^{2} + 101\cdot 163^{3} + 90\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 value |
| $1$ | $1$ | $()$ | $3$ |
| $3$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $6$ | $2$ | $(1,2)$ | $-1$ |
| $8$ | $3$ | $(1,2,3)$ | $0$ |
| $6$ | $4$ | $(1,2,3,4)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
