Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 3\cdot 43 + 24\cdot 43^{2} + 5\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 13\cdot 43 + 19\cdot 43^{2} + 5\cdot 43^{3} + 21\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 14\cdot 43 + 26\cdot 43^{2} + 32\cdot 43^{3} + 5\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 12\cdot 43 + 2\cdot 43^{2} + 38\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 33 + 42\cdot 43 + 13\cdot 43^{2} + 4\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,4,3,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,2,4,3,5)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,4,5,2,3)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,3,2,5,4)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,5,3,4,2)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
