Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 14\cdot 31 + 23\cdot 31^{2} + 17\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 8\cdot 31 + 6\cdot 31^{2} + 28\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 26\cdot 31 + 16\cdot 31^{2} + 11\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 8 + 3\cdot 31 + 23\cdot 31^{2} + 3\cdot 31^{3} + 2\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 13 + 27\cdot 31 + 17\cdot 31^{2} + 30\cdot 31^{3} + 17\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 15 + 18\cdot 31 + 29\cdot 31^{2} + 28\cdot 31^{3} + 30\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 18 + 12\cdot 31 + 26\cdot 31^{2} + 5\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 28 + 13\cdot 31 + 11\cdot 31^{2} + 28\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,6,8,3,2,4,7)$ |
| $(1,6,3,4)(2,7,5,8)$ |
| $(2,7)(4,6)(5,8)$ |
| $(1,3)(2,5)(4,6)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,3)(2,5)(4,6)(7,8)$ |
$-2$ |
$-2$ |
| $4$ |
$2$ |
$(2,7)(4,6)(5,8)$ |
$0$ |
$0$ |
| $2$ |
$4$ |
$(1,6,3,4)(2,7,5,8)$ |
$0$ |
$0$ |
| $4$ |
$4$ |
$(1,8,3,7)(2,6,5,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,6,8,3,2,4,7)$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ |
$8$ |
$(1,2,6,7,3,5,4,8)$ |
$\zeta_{8}^{3} + \zeta_{8}$ |
$-\zeta_{8}^{3} - \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
