Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 523 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 89 + 63\cdot 523 + 158\cdot 523^{2} + 424\cdot 523^{3} + 195\cdot 523^{4} +O\left(523^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 221 + 98\cdot 523 + 300\cdot 523^{2} + 7\cdot 523^{3} + 82\cdot 523^{4} +O\left(523^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 361 + 25\cdot 523 + 511\cdot 523^{2} + 100\cdot 523^{3} + 462\cdot 523^{4} +O\left(523^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 437 + 335\cdot 523 + 95\cdot 523^{2} + 349\cdot 523^{3} + 94\cdot 523^{4} +O\left(523^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 462 + 522\cdot 523 + 503\cdot 523^{2} + 163\cdot 523^{3} + 211\cdot 523^{4} +O\left(523^{ 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 |
| $1$ | $1$ | $()$ | $3$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $20$ | $3$ | $(1,2,3)$ | $0$ |
| $12$ | $5$ | $(1,2,3,4,5)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2}$ |
| $12$ | $5$ | $(1,3,4,5,2)$ | $\zeta_{5}^{3} + \zeta_{5}^{2} + 1$ |
The blue line marks the conjugacy class containing complex conjugation.
