Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 8.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 19\cdot 31 + 23\cdot 31^{2} + 20\cdot 31^{3} + 18\cdot 31^{4} + 26\cdot 31^{5} + 28\cdot 31^{6} + 9\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 14\cdot 31 + 28\cdot 31^{2} + 28\cdot 31^{3} + 14\cdot 31^{4} + 31^{5} + 7\cdot 31^{6} + 19\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 7\cdot 31 + 31^{2} + 13\cdot 31^{3} + 13\cdot 31^{4} + 10\cdot 31^{5} + 23\cdot 31^{6} + 24\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 15 + 30\cdot 31 + 6\cdot 31^{2} + 13\cdot 31^{4} + 28\cdot 31^{5} + 27\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 16 + 24\cdot 31^{2} + 30\cdot 31^{3} + 17\cdot 31^{4} + 2\cdot 31^{5} + 30\cdot 31^{6} + 3\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 23\cdot 31 + 29\cdot 31^{2} + 17\cdot 31^{3} + 17\cdot 31^{4} + 20\cdot 31^{5} + 7\cdot 31^{6} + 6\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 16\cdot 31 + 2\cdot 31^{2} + 2\cdot 31^{3} + 16\cdot 31^{4} + 29\cdot 31^{5} + 23\cdot 31^{6} + 11\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 + 11\cdot 31 + 7\cdot 31^{2} + 10\cdot 31^{3} + 12\cdot 31^{4} + 4\cdot 31^{5} + 2\cdot 31^{6} + 21\cdot 31^{7} +O\left(31^{ 8 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(1,5,7,6,8,4,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
| $1$ |
$8$ |
$(1,5,7,6,8,4,2,3)$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,6,2,5,8,3,7,4)$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
| $1$ |
$8$ |
$(1,4,7,3,8,5,2,6)$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,3,2,4,8,6,7,5)$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
