Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 283 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 25 + 80\cdot 283 + 155\cdot 283^{2} + 69\cdot 283^{3} + 145\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 82 + 255\cdot 283 + 46\cdot 283^{2} + 254\cdot 283^{3} + 259\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 201 + 27\cdot 283 + 236\cdot 283^{2} + 28\cdot 283^{3} + 23\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 258 + 202\cdot 283 + 127\cdot 283^{2} + 213\cdot 283^{3} + 137\cdot 283^{4} +O\left(283^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
