Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 35\cdot 47 + 41\cdot 47^{2} + 9\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 10\cdot 47 + 41\cdot 47^{2} + 8\cdot 47^{3} + 34\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 5 + 5\cdot 47 + 8\cdot 47^{2} + 19\cdot 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 30 + 34\cdot 47 + 4\cdot 47^{2} + 31\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 34\cdot 47 + 28\cdot 47^{2} + 5\cdot 47^{3} + 13\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 38 + 9\cdot 47 + 42\cdot 47^{2} + 34\cdot 47^{3} + 33\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 40\cdot 47 + 32\cdot 47^{2} + 19\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 17\cdot 47 + 35\cdot 47^{2} + 11\cdot 47^{3} + 37\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,6)(4,7)(5,8)$ |
| $(1,2,7,8)(3,5,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,7)(2,8)(3,4)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,6)(4,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ |
| $2$ | $4$ | $(1,2,7,8)(3,5,4,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
