Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 563 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 179 + 18\cdot 563 + 551\cdot 563^{2} + 304\cdot 563^{3} + 453\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 326 + 84\cdot 563 + 490\cdot 563^{2} + 130\cdot 563^{3} + 130\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 334 + 291\cdot 563 + 49\cdot 563^{2} + 237\cdot 563^{3} + 54\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 422 + 138\cdot 563 + 177\cdot 563^{2} + 436\cdot 563^{3} + 317\cdot 563^{4} +O\left(563^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 429 + 29\cdot 563 + 421\cdot 563^{2} + 16\cdot 563^{3} + 170\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.
