Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 31 + 10\cdot 31^{2} + 17\cdot 31^{3} + 31^{4} + 26\cdot 31^{5} + 21\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 28\cdot 31 + 26\cdot 31^{2} + 12\cdot 31^{3} + 18\cdot 31^{5} + 19\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 3\cdot 31 + 21\cdot 31^{2} + 4\cdot 31^{3} + 15\cdot 31^{4} + 19\cdot 31^{5} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 12 + 21\cdot 31 + 27\cdot 31^{2} + 3\cdot 31^{3} + 17\cdot 31^{4} + 31^{5} + 11\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 19 + 9\cdot 31 + 3\cdot 31^{2} + 27\cdot 31^{3} + 13\cdot 31^{4} + 29\cdot 31^{5} + 19\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 27\cdot 31 + 9\cdot 31^{2} + 26\cdot 31^{3} + 15\cdot 31^{4} + 11\cdot 31^{5} + 30\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 2\cdot 31 + 4\cdot 31^{2} + 18\cdot 31^{3} + 30\cdot 31^{4} + 12\cdot 31^{5} + 11\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 + 9\cdot 31 + 20\cdot 31^{2} + 13\cdot 31^{3} + 29\cdot 31^{4} + 4\cdot 31^{5} + 9\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,5)(4,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
