Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 31\cdot 41 + 39\cdot 41^{2} + 21\cdot 41^{3} + 11\cdot 41^{4} + 8\cdot 41^{5} + 38\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 5\cdot 41 + 34\cdot 41^{2} + 41^{3} + 35\cdot 41^{4} + 36\cdot 41^{5} + 26\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 33 + 35\cdot 41 + 6\cdot 41^{2} + 39\cdot 41^{3} + 5\cdot 41^{4} + 4\cdot 41^{5} + 14\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 35 + 9\cdot 41 + 41^{2} + 19\cdot 41^{3} + 29\cdot 41^{4} + 32\cdot 41^{5} + 2\cdot 41^{6} +O\left(41^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
