Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 9\cdot 31 + 8\cdot 31^{2} + 30\cdot 31^{3} + 2\cdot 31^{4} + 21\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 6\cdot 31 + 18\cdot 31^{2} + 7\cdot 31^{3} + 9\cdot 31^{4} + 6\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 21 + 25\cdot 31 + 19\cdot 31^{2} + 18\cdot 31^{3} + 4\cdot 31^{4} + 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 20\cdot 31 + 15\cdot 31^{2} + 5\cdot 31^{3} + 14\cdot 31^{4} + 2\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(1,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,4)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(2,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,3,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
