Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 7 + 28\cdot 31 + 31^{2} + 10\cdot 31^{3} + 7\cdot 31^{4} + 17\cdot 31^{5} + 28\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 4\cdot 31 + 30\cdot 31^{2} + 3\cdot 31^{3} + 25\cdot 31^{4} + 20\cdot 31^{5} + 20\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 19 + 26\cdot 31 + 27\cdot 31^{3} + 5\cdot 31^{4} + 10\cdot 31^{5} + 10\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 24 + 2\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 23\cdot 31^{4} + 13\cdot 31^{5} + 2\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 4 }$
| Cycle notation |
| $(1,2)(3,4)$ |
| $(2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 4 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)$ |
$0$ |
| $2$ |
$2$ |
$(1,4)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,2)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
