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