Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 223 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 63 + 34\cdot 223 + 103\cdot 223^{2} + 201\cdot 223^{3} + 64\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 82 + 115\cdot 223 + 182\cdot 223^{2} + 75\cdot 223^{3} + 97\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 141 + 107\cdot 223 + 40\cdot 223^{2} + 147\cdot 223^{3} + 125\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 160 + 188\cdot 223 + 119\cdot 223^{2} + 21\cdot 223^{3} + 158\cdot 223^{4} +O\left(223^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
