Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 54 + 21\cdot 257 + 142\cdot 257^{2} + 138\cdot 257^{3} + 153\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 90 + 102\cdot 257 + 212\cdot 257^{2} + 217\cdot 257^{3} + 210\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 107 + 6\cdot 257 + 204\cdot 257^{2} + 210\cdot 257^{3} + 181\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 117 + 113\cdot 257 + 152\cdot 257^{2} + 253\cdot 257^{3} + 223\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 148 + 13\cdot 257 + 60\cdot 257^{2} + 207\cdot 257^{3} +O\left(257^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3)$ |
| $(3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
