Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 9\cdot 31 + 21\cdot 31^{2} + 31^{3} + 4\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 3 + 5\cdot 31 + 7\cdot 31^{2} + 24\cdot 31^{3} + 7\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 22\cdot 31 + 8\cdot 31^{2} + 16\cdot 31^{3} + 14\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 10 + 10\cdot 31 + 16\cdot 31^{2} + 17\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 12 + 20\cdot 31 + 15\cdot 31^{2} + 26\cdot 31^{3} + 21\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 14 + 22\cdot 31 + 25\cdot 31^{2} + 6\cdot 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 21 + 8\cdot 31 + 10\cdot 31^{2} + 29\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 24 + 25\cdot 31 + 18\cdot 31^{2} + 31^{3} + 3\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,6)$ |
| $(1,6,3,2)(4,8,5,7)$ |
| $(1,8,3,7)(2,5,6,4)$ |
| $(1,3)(2,6)(4,5)(7,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,6)(4,5)(7,8)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)$ | $0$ |
| $2$ | $2$ | $(1,7)(2,4)(3,8)(5,6)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,2,3,6)(4,8,5,7)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,3,2)(4,7,5,8)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,3,7)(2,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,6,3,2)(4,8,5,7)$ | $0$ |
| $2$ | $4$ | $(1,5,3,4)(2,7,6,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
