Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 5 + 18\cdot 31 + 30\cdot 31^{2} + 27\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 19\cdot 31 + 12\cdot 31^{2} + 3\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 11\cdot 31 + 27\cdot 31^{2} + 20\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 29 + 12\cdot 31 + 22\cdot 31^{2} + 9\cdot 31^{3} + 9\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3,2,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-1$ |
| $1$ | $4$ | $(1,3,2,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,2,3)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
