Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 443 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 65 + 366\cdot 443 + 206\cdot 443^{2} + 193\cdot 443^{3} + 240\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 86 + 6\cdot 443 + 121\cdot 443^{2} + 222\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 157 + 215\cdot 443 + 325\cdot 443^{2} + 66\cdot 443^{3} + 392\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 203 + 158\cdot 443 + 430\cdot 443^{2} + 105\cdot 443^{3} + 27\cdot 443^{4} +O\left(443^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 376 + 139\cdot 443 + 245\cdot 443^{2} + 76\cdot 443^{3} + 4\cdot 443^{4} +O\left(443^{ 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 |
| $1$ | $1$ | $()$ | $4$ |
| $10$ | $2$ | $(1,2)$ | $2$ |
| $15$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $20$ | $3$ | $(1,2,3)$ | $1$ |
| $30$ | $4$ | $(1,2,3,4)$ | $0$ |
| $24$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $20$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.
