Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 11 + 18\cdot 43^{2} + 43^{3} + 37\cdot 43^{4} + 3\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 33\cdot 43 + 19\cdot 43^{2} + 22\cdot 43^{3} + 25\cdot 43^{4} + 11\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 + 9\cdot 43 + 6\cdot 43^{2} + 19\cdot 43^{3} + 12\cdot 43^{4} + 41\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 32 + 42\cdot 43 + 41\cdot 43^{2} + 42\cdot 43^{3} + 10\cdot 43^{4} + 29\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,3,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
