Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 50 + 33\cdot 283 + 161\cdot 283^{2} + 62\cdot 283^{3} + 6\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 107 + 208\cdot 283 + 52\cdot 283^{2} + 247\cdot 283^{3} + 120\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 152 + 277\cdot 283 + 74\cdot 283^{2} + 249\cdot 283^{3} + 16\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 259 + 46\cdot 283 + 277\cdot 283^{2} + 6\cdot 283^{3} + 139\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,3)(2,4)$ |
| $(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,2)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,2,3)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
