Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 62 + 490\cdot 563 + 556\cdot 563^{2} + 137\cdot 563^{3} + 481\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 92 + 96\cdot 563 + 22\cdot 563^{2} + 500\cdot 563^{3} + 453\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 119 + 419\cdot 563 + 335\cdot 563^{2} + 36\cdot 563^{3} + 335\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 322 + 504\cdot 563 + 238\cdot 563^{2} + 528\cdot 563^{3} + 182\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 531 + 178\cdot 563 + 535\cdot 563^{2} + 485\cdot 563^{3} + 235\cdot 563^{4} +O\left(563^{ 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 value |
| $1$ | $1$ | $()$ | $4$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
