Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 13\cdot 43 + 4\cdot 43^{2} + 33\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 18 + 19\cdot 43 + 35\cdot 43^{2} + 15\cdot 43^{3} + 9\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 5\cdot 43 + 5\cdot 43^{2} + 11\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 12\cdot 43 + 38\cdot 43^{2} + 27\cdot 43^{3} + 10\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 35\cdot 43 + 2\cdot 43^{2} + 41\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,5,3,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$5$ |
$(1,2,5,3,4)$ |
$\zeta_{5}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}^{3}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ |
$5$ |
$(1,5,4,2,3)$ |
$\zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}$ |
$\zeta_{5}^{3}$ |
| $1$ |
$5$ |
$(1,3,2,4,5)$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{2}$ |
| $1$ |
$5$ |
$(1,4,3,5,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.
