Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 227 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 18\cdot 227 + 152\cdot 227^{2} + 79\cdot 227^{3} + 106\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 89 + 219\cdot 227 + 112\cdot 227^{2} + 20\cdot 227^{3} + 47\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 154 + 42\cdot 227 + 50\cdot 227^{2} + 123\cdot 227^{3} + 108\cdot 227^{4} +O\left(227^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 208 + 173\cdot 227 + 138\cdot 227^{2} + 3\cdot 227^{3} + 192\cdot 227^{4} +O\left(227^{ 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.
