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