Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 31 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 17\cdot 31 + 9\cdot 31^{2} + 11\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 8 + 26\cdot 31 + 10\cdot 31^{2} + 16\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 7\cdot 31 + 13\cdot 31^{3} + 16\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 30\cdot 31 + 20\cdot 31^{2} + 24\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 23\cdot 31 + 10\cdot 31^{3} + 25\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 14\cdot 31 + 2\cdot 31^{2} + 17\cdot 31^{3} + 4\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 23\cdot 31 + 16\cdot 31^{2} + 8\cdot 31^{3} + 31^{4} +O\left(31^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 30 + 10\cdot 31 + 23\cdot 31^{3} + 24\cdot 31^{4} +O\left(31^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,7,3,6,8,4,5)$ |
| $(1,7,6,4)(2,5,8,3)$ |
| $(2,8)(3,5)$ |
| $(1,6)(4,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,6)(2,8)(3,5)(4,7)$ | $-2$ |
| $2$ | $2$ | $(1,6)(4,7)$ | $0$ |
| $1$ | $4$ | $(1,7,6,4)(2,3,8,5)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,4,6,7)(2,5,8,3)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,6,4)(2,5,8,3)$ | $0$ |
| $2$ | $8$ | $(1,2,7,3,6,8,4,5)$ | $0$ |
| $2$ | $8$ | $(1,3,4,2,6,5,7,8)$ | $0$ |
| $2$ | $8$ | $(1,8,4,3,6,2,7,5)$ | $0$ |
| $2$ | $8$ | $(1,3,7,8,6,5,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
