Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 318 + 227\cdot 563 + 134\cdot 563^{2} + 32\cdot 563^{3} + 43\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 441 + 120\cdot 563 + 61\cdot 563^{2} + 449\cdot 563^{3} + 91\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 460 + 156\cdot 563 + 321\cdot 563^{2} + 68\cdot 563^{3} + 506\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 485 + 416\cdot 563 + 561\cdot 563^{2} + 525\cdot 563^{3} + 313\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 550 + 203\cdot 563 + 47\cdot 563^{2} + 50\cdot 563^{3} + 171\cdot 563^{4} +O\left(563^{ 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.
