Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 12\cdot 31 + 12\cdot 31^{2} + 21\cdot 31^{4} + 26\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 11 + 30\cdot 31 + 4\cdot 31^{2} + 21\cdot 31^{3} + 18\cdot 31^{4} + 24\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 20\cdot 31 + 24\cdot 31^{2} + 10\cdot 31^{3} + 28\cdot 31^{4} + 20\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 17 + 2\cdot 31 + 14\cdot 31^{2} + 18\cdot 31^{3} + 10\cdot 31^{4} + 9\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 27\cdot 31 + 5\cdot 31^{2} + 11\cdot 31^{3} + 14\cdot 31^{4} + 11\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,2,3,5,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$5$ |
$(1,2,3,5,4)$ |
$\zeta_{5}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}^{3}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ |
$5$ |
$(1,3,4,2,5)$ |
$\zeta_{5}^{2}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}$ |
$\zeta_{5}^{3}$ |
| $1$ |
$5$ |
$(1,5,2,4,3)$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{2}$ |
| $1$ |
$5$ |
$(1,4,5,3,2)$ |
$-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
$\zeta_{5}^{3}$ |
$\zeta_{5}^{2}$ |
$\zeta_{5}$ |
The blue line marks the conjugacy class containing complex conjugation.
