Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 557 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 14 + 251\cdot 557 + 288\cdot 557^{2} + 375\cdot 557^{3} + 218\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 235 + 435\cdot 557 + 247\cdot 557^{2} + 321\cdot 557^{3} + 239\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 375 + 224\cdot 557 + 116\cdot 557^{2} + 75\cdot 557^{3} + 92\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 506 + 262\cdot 557 + 255\cdot 557^{2} + 489\cdot 557^{3} + 304\cdot 557^{4} +O\left(557^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 542 + 496\cdot 557 + 205\cdot 557^{2} + 409\cdot 557^{3} + 258\cdot 557^{4} +O\left(557^{ 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.
