Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 24\cdot 31 + 19\cdot 31^{2} + 5\cdot 31^{3} + 24\cdot 31^{4} + 25\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 24\cdot 31 + 30\cdot 31^{2} + 2\cdot 31^{3} + 2\cdot 31^{4} + 29\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 8 + 21\cdot 31 + 22\cdot 31^{2} + 16\cdot 31^{3} + 8\cdot 31^{4} + 26\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 8\cdot 31 + 10\cdot 31^{2} + 15\cdot 31^{3} + 26\cdot 31^{4} + 5\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 18 + 22\cdot 31 + 20\cdot 31^{2} + 15\cdot 31^{3} + 4\cdot 31^{4} + 25\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 23 + 9\cdot 31 + 8\cdot 31^{2} + 14\cdot 31^{3} + 22\cdot 31^{4} + 4\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 6\cdot 31 + 28\cdot 31^{3} + 28\cdot 31^{4} + 31^{5} +O\left(31^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 + 6\cdot 31 + 11\cdot 31^{2} + 25\cdot 31^{3} + 6\cdot 31^{4} + 5\cdot 31^{5} +O\left(31^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7)(2,8)(3,5)(4,6)$ |
| $(1,4,8,5)(2,3,7,6)$ |
| $(1,7,8,2)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $2$ | $(2,7)(3,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,8)(3,5)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
