Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 20\cdot 31 + 21\cdot 31^{2} + 28\cdot 31^{3} + 31^{4} + 11\cdot 31^{5} + 24\cdot 31^{6} + 13\cdot 31^{7} + 18\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 10\cdot 31^{2} + 28\cdot 31^{3} + 6\cdot 31^{4} + 29\cdot 31^{5} + 19\cdot 31^{6} + 21\cdot 31^{7} + 19\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 3 + 17\cdot 31 + 24\cdot 31^{2} + 4\cdot 31^{3} + 8\cdot 31^{4} + 6\cdot 31^{5} + 4\cdot 31^{6} + 19\cdot 31^{7} + 4\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 31 + 21\cdot 31^{2} + 28\cdot 31^{3} + 9\cdot 31^{4} + 24\cdot 31^{5} + 10\cdot 31^{7} + 15\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 15 + 30\cdot 31 + 23\cdot 31^{2} + 28\cdot 31^{3} + 6\cdot 31^{4} + 14\cdot 31^{5} + 27\cdot 31^{6} + 28\cdot 31^{7} + 25\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 17 + 23\cdot 31 + 16\cdot 31^{2} + 6\cdot 31^{3} + 13\cdot 31^{4} + 22\cdot 31^{5} + 12\cdot 31^{6} + 11\cdot 31^{7} + 28\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 25 + 20\cdot 31 + 15\cdot 31^{2} + 8\cdot 31^{4} + 21\cdot 31^{5} + 20\cdot 31^{6} + 24\cdot 31^{7} + 19\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 26 + 10\cdot 31 + 21\cdot 31^{2} + 28\cdot 31^{3} + 6\cdot 31^{4} + 26\cdot 31^{5} + 13\cdot 31^{6} + 25\cdot 31^{7} + 22\cdot 31^{8} +O\left(31^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,6,4)(2,5,3,7)$ |
| $(2,3)(5,7)$ |
| $(4,8)(5,7)$ |
| $(1,7,8,3,6,5,4,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,6)(2,3)(4,8)(5,7)$ |
$-4$ |
| $2$ |
$2$ |
$(2,3)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(4,8)(5,7)$ |
$0$ |
| $4$ |
$2$ |
$(1,8)(2,5)(3,7)(4,6)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,6,4)(2,5,3,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,8,6,4)(2,7,3,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,8,3,6,5,4,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,4,7,6,2,8,5)$ |
$0$ |
| $4$ |
$8$ |
$(1,7,4,3,6,5,8,2)$ |
$0$ |
| $4$ |
$8$ |
$(1,3,8,7,6,2,4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
