Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
$r_{ 1 }$ |
$=$ |
$ 16 + 27\cdot 41 + 34\cdot 41^{2} + 6\cdot 41^{3} + 17\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
$r_{ 2 }$ |
$=$ |
$ 19 + 6\cdot 41 + 39\cdot 41^{2} + 12\cdot 41^{3} + 38\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
$r_{ 3 }$ |
$=$ |
$ 22 + 34\cdot 41 + 41^{2} + 28\cdot 41^{3} + 2\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
$r_{ 4 }$ |
$=$ |
$ 25 + 13\cdot 41 + 6\cdot 41^{2} + 34\cdot 41^{3} + 23\cdot 41^{4} +O\left(41^{ 5 }\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 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.