Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 5\cdot 31 + 8\cdot 31^{2} + 6\cdot 31^{3} + 26\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 6\cdot 31 + 22\cdot 31^{2} + 26\cdot 31^{3} + 8\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 11 + 6\cdot 31 + 2\cdot 31^{2} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 8\cdot 31 + 26\cdot 31^{2} + 7\cdot 31^{3} + 6\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 4\cdot 31 + 3\cdot 31^{2} + 21\cdot 31^{3} + 19\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 5 }$
| Cycle notation |
| $(1,3,5,4,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 5 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $5$ | $(1,3,5,4,2)$ | $\zeta_{5}^{2}$ |
| $1$ | $5$ | $(1,5,2,3,4)$ | $-\zeta_{5}^{3} - \zeta_{5}^{2} - \zeta_{5} - 1$ |
| $1$ | $5$ | $(1,4,3,2,5)$ | $\zeta_{5}$ |
| $1$ | $5$ | $(1,2,4,5,3)$ | $\zeta_{5}^{3}$ |
The blue line marks the conjugacy class containing complex conjugation.
