Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 257 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 112\cdot 257 + 39\cdot 257^{2} + 171\cdot 257^{3} + 199\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 20 + 213\cdot 257 + 152\cdot 257^{2} + 58\cdot 257^{3} + 44\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 55 + 106\cdot 257 + 180\cdot 257^{2} + 38\cdot 257^{3} + 134\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 181 + 87\cdot 257 + 92\cdot 257^{2} + 133\cdot 257^{3} + 165\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 253 + 251\cdot 257 + 48\cdot 257^{2} + 112\cdot 257^{3} + 227\cdot 257^{4} +O\left(257^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2)$ |
| $(1,2,3,4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $5$ |
| $10$ | $2$ | $(1,2)$ | $1$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $20$ | $3$ | $(1,2,3)$ | $-1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $1$ |
The blue line marks the conjugacy class containing complex conjugation.
