Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 17 + 27\cdot 31 + 12\cdot 31^{2} + 15\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 22 + 19\cdot 31 + 28\cdot 31^{2} + 18\cdot 31^{3} + 22\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 27 + 9\cdot 31 + 30\cdot 31^{2} + 15\cdot 31^{3} + 5\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 28 + 4\cdot 31 + 21\cdot 31^{2} + 11\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 |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $1$ | $4$ | $(1,2,3,4)$ | $-\zeta_{4}$ |
| $1$ | $4$ | $(1,4,3,2)$ | $\zeta_{4}$ |
The blue line marks the conjugacy class containing complex conjugation.
