Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2\cdot 41 + 5\cdot 41^{2} + 26\cdot 41^{3} + 19\cdot 41^{4} + 17\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 30\cdot 41 + 31\cdot 41^{2} + 3\cdot 41^{3} + 6\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 20\cdot 41 + 2\cdot 41^{2} + 19\cdot 41^{3} + 18\cdot 41^{4} + 10\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 33 + 29\cdot 41 + 41^{2} + 33\cdot 41^{3} + 37\cdot 41^{4} + 2\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)$ |
| $(1,2)(3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
