Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 23\cdot 31 + 22\cdot 31^{2} + 31^{3} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 29\cdot 31 + 3\cdot 31^{2} + 8\cdot 31^{3} + 20\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 10\cdot 31 + 19\cdot 31^{2} + 2\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 17\cdot 31 + 27\cdot 31^{2} + 28\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 28 + 12\cdot 31 + 19\cdot 31^{2} + 20\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 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.
