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