Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 41\cdot 43 + 5\cdot 43^{2} + 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 19\cdot 43 + 20\cdot 43^{2} + 31\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 42\cdot 43 + 34\cdot 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 34 + 26\cdot 43 + 15\cdot 43^{2} + 7\cdot 43^{3} + 20\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,4)(2,3)$ |
| $(1,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-1$ |
| $1$ | $4$ | $(1,2,4,3)$ | $\zeta_{4}$ |
| $1$ | $4$ | $(1,3,4,2)$ | $-\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
