Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 15 + 403\cdot 563 + 196\cdot 563^{2} + 519\cdot 563^{3} + 405\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 59 + 293\cdot 563 + 310\cdot 563^{2} + 276\cdot 563^{3} + 183\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 130 + 285\cdot 563 + 182\cdot 563^{2} + 119\cdot 563^{3} + 83\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 385 + 201\cdot 563 + 365\cdot 563^{2} + 44\cdot 563^{3} + 405\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 538 + 505\cdot 563 + 70\cdot 563^{2} + 166\cdot 563^{3} + 48\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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
