Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 547 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 91 + 452\cdot 547 + 370\cdot 547^{2} + 433\cdot 547^{3} + 238\cdot 547^{4} +O\left(547^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 351 + 192\cdot 547 + 518\cdot 547^{2} + 440\cdot 547^{3} + 467\cdot 547^{4} +O\left(547^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 371 + 134\cdot 547 + 412\cdot 547^{2} + 141\cdot 547^{3} + 402\cdot 547^{4} +O\left(547^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 400 + 62\cdot 547 + 211\cdot 547^{2} + 524\cdot 547^{3} + 435\cdot 547^{4} +O\left(547^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 429 + 251\cdot 547 + 128\cdot 547^{2} + 100\cdot 547^{3} + 96\cdot 547^{4} +O\left(547^{ 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 values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$3$ |
$3$ |
| $15$ |
$2$ |
$(1,2)(3,4)$ |
$-1$ |
$-1$ |
| $20$ |
$3$ |
$(1,2,3)$ |
$0$ |
$0$ |
| $12$ |
$5$ |
$(1,2,3,4,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
| $12$ |
$5$ |
$(1,3,4,5,2)$ |
$\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
The blue line marks the conjugacy class containing complex conjugation.
