Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 229 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 32 + 87\cdot 229 + 6\cdot 229^{2} + 83\cdot 229^{3} + 182\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 58 + 48\cdot 229 + 182\cdot 229^{2} + 126\cdot 229^{3} + 209\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 63 + 147\cdot 229 + 143\cdot 229^{2} + 161\cdot 229^{3} + 3\cdot 229^{4} +O\left(229^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 78 + 175\cdot 229 + 125\cdot 229^{2} + 86\cdot 229^{3} + 62\cdot 229^{4} +O\left(229^{ 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.
