Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 443 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 52 + 114\cdot 443 + 237\cdot 443^{2} + 182\cdot 443^{3} + 316\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 123 + 169\cdot 443 + 115\cdot 443^{2} + 312\cdot 443^{3} + 425\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 180 + 331\cdot 443 + 321\cdot 443^{2} + 174\cdot 443^{3} + 391\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 200 + 294\cdot 443 + 244\cdot 443^{2} + 49\cdot 443^{3} + 96\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 333 + 419\cdot 443 + 409\cdot 443^{2} + 166\cdot 443^{3} + 99\cdot 443^{4} +O\left(443^{ 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.
