Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 18\cdot 31 + 30\cdot 31^{2} + 23\cdot 31^{3} + 13\cdot 31^{4} + 16\cdot 31^{5} +O\left(31^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 11\cdot 31 + 26\cdot 31^{2} + 31^{3} + 6\cdot 31^{4} + 28\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 9 + 18\cdot 31 + 29\cdot 31^{2} + 20\cdot 31^{3} + 20\cdot 31^{4} + 27\cdot 31^{5} + 5\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 14 + 2\cdot 31 + 25\cdot 31^{3} + 29\cdot 31^{4} + 5\cdot 31^{5} + 11\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 17 + 28\cdot 31 + 30\cdot 31^{2} + 5\cdot 31^{3} + 31^{4} + 25\cdot 31^{5} + 19\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 22 + 12\cdot 31 + 31^{2} + 10\cdot 31^{3} + 10\cdot 31^{4} + 3\cdot 31^{5} + 25\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 24 + 19\cdot 31 + 4\cdot 31^{2} + 29\cdot 31^{3} + 24\cdot 31^{4} + 30\cdot 31^{5} + 2\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 + 12\cdot 31 + 7\cdot 31^{3} + 17\cdot 31^{4} + 14\cdot 31^{5} + 30\cdot 31^{6} +O\left(31^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,4,6,7,8,5,3,2)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
$c3$ |
$c4$ |
| $1$ |
$1$ |
$()$ |
$1$ |
$1$ |
$1$ |
$1$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-1$ |
$-1$ |
$-1$ |
$-1$ |
| $1$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
| $1$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
$-\zeta_{8}^{2}$ |
$\zeta_{8}^{2}$ |
| $1$ |
$8$ |
$(1,4,6,7,8,5,3,2)$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,7,3,4,8,2,6,5)$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
| $1$ |
$8$ |
$(1,5,6,2,8,4,3,7)$ |
$-\zeta_{8}$ |
$-\zeta_{8}^{3}$ |
$\zeta_{8}$ |
$\zeta_{8}^{3}$ |
| $1$ |
$8$ |
$(1,2,3,5,8,7,6,4)$ |
$-\zeta_{8}^{3}$ |
$-\zeta_{8}$ |
$\zeta_{8}^{3}$ |
$\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
