Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 23\cdot 31 + 20\cdot 31^{2} + 25\cdot 31^{3} + 23\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 15\cdot 31 + 20\cdot 31^{2} + 24\cdot 31^{3} + 28\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 16\cdot 31 + 8\cdot 31^{2} + 8\cdot 31^{3} + 4\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 14\cdot 31 + 4\cdot 31^{2} + 25\cdot 31^{3} + 12\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 16\cdot 31 + 26\cdot 31^{2} + 5\cdot 31^{3} + 18\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 14\cdot 31 + 22\cdot 31^{2} + 22\cdot 31^{3} + 26\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 + 15\cdot 31 + 10\cdot 31^{2} + 6\cdot 31^{3} + 2\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 27 + 7\cdot 31 + 10\cdot 31^{2} + 5\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
