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