Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 33\cdot 47 + 33\cdot 47^{2} + 46\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 28\cdot 47 + 29\cdot 47^{2} + 14\cdot 47^{3} + 29\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 25\cdot 47 + 15\cdot 47^{2} + 46\cdot 47^{3} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 34\cdot 47 + 2\cdot 47^{2} + 13\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 12\cdot 47 + 44\cdot 47^{2} + 33\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 21\cdot 47 + 31\cdot 47^{2} + 46\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 18\cdot 47 + 17\cdot 47^{2} + 32\cdot 47^{3} + 17\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 13\cdot 47 + 13\cdot 47^{2} + 38\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,4,6,5)$ |
| $(1,3,2,5,8,6,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $1$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-1$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-\zeta_{8}^{2}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $\zeta_{8}^{2}$ |
| $1$ | $8$ | $(1,3,2,5,8,6,7,4)$ | $\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $\zeta_{8}$ |
| $1$ | $8$ | $(1,6,2,4,8,3,7,5)$ | $-\zeta_{8}^{3}$ |
| $1$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $-\zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.
