Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 41\cdot 47 + 14\cdot 47^{2} + 33\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 35\cdot 47 + 31\cdot 47^{2} + 27\cdot 47^{3} + 19\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 4\cdot 47 + 24\cdot 47^{2} + 20\cdot 47^{3} + 18\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 7\cdot 47 + 24\cdot 47^{2} + 46\cdot 47^{3} + 46\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 + 4\cdot 47 + 14\cdot 47^{2} + 42\cdot 47^{3} + 39\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 24\cdot 47 + 41\cdot 47^{2} + 47^{3} + 9\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 41 + 23\cdot 47 + 34\cdot 47^{2} + 6\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 46\cdot 47 + 2\cdot 47^{2} + 9\cdot 47^{3} + 38\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3,4,8)(2,5,7,6)$ |
| $(1,2)(3,6)(4,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,7)(3,8)(5,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,6)(7,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,4,8)(2,5,7,6)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
