Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 12\cdot 47 + 13\cdot 47^{2} + 17\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 31\cdot 47 + 39\cdot 47^{2} + 6\cdot 47^{3} + 40\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 15\cdot 47 + 20\cdot 47^{2} + 11\cdot 47^{3} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 37\cdot 47 + 3\cdot 47^{2} + 10\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 38 + 44\cdot 47 + 16\cdot 47^{2} + 47^{3} + 31\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,5,2,4,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$5$ |
$(1,5,2,4,3)$ |
$\zeta_{5}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}^{3}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ |
$5$ |
$(1,2,3,5,4)$ |
$\zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}$ |
$\zeta_{5}^{3}$ |
| $1$ |
$5$ |
$(1,4,5,3,2)$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{2}$ |
| $1$ |
$5$ |
$(1,3,4,2,5)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.
