Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 3\cdot 31 + 12\cdot 31^{2} + 23\cdot 31^{3} + 26\cdot 31^{4} + 5\cdot 31^{5} + 22\cdot 31^{6} + 13\cdot 31^{7} + 24\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 11\cdot 31 + 6\cdot 31^{2} + 13\cdot 31^{3} + 12\cdot 31^{4} + 24\cdot 31^{5} + 18\cdot 31^{6} + 30\cdot 31^{7} + 4\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 26\cdot 31 + 5\cdot 31^{2} + 18\cdot 31^{3} + 17\cdot 31^{4} + 15\cdot 31^{5} + 22\cdot 31^{6} + 31^{7} + 3\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 21\cdot 31 + 6\cdot 31^{2} + 7\cdot 31^{3} + 5\cdot 31^{4} + 16\cdot 31^{5} + 29\cdot 31^{6} + 15\cdot 31^{7} + 29\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 30\cdot 31 + 25\cdot 31^{2} + 18\cdot 31^{3} + 13\cdot 31^{4} + 17\cdot 31^{5} + 27\cdot 31^{6} + 23\cdot 31^{7} + 20\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 21 + 13\cdot 31 + 11\cdot 31^{2} + 2\cdot 31^{3} + 3\cdot 31^{4} + 17\cdot 31^{5} + 22\cdot 31^{6} + 31^{7} + 5\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 26 + 6\cdot 31 + 17\cdot 31^{2} + 12\cdot 31^{3} + 16\cdot 31^{4} + 22\cdot 31^{5} + 13\cdot 31^{6} + 8\cdot 31^{7} + 18\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 29 + 10\cdot 31 + 7\cdot 31^{2} + 28\cdot 31^{3} + 28\cdot 31^{4} + 4\cdot 31^{5} + 29\cdot 31^{6} + 27\cdot 31^{7} + 17\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,4)(5,6)(7,8)$ |
| $(1,5,4,7)(2,6,3,8)$ |
| $(1,2)(3,4)(5,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,7)(6,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,6)(2,5)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,5,4,7)(2,6,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
