Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 8\cdot 47 + 14\cdot 47^{2} + 21\cdot 47^{3} + 40\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 45\cdot 47 + 21\cdot 47^{2} + 20\cdot 47^{3} + 22\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 12 + 12\cdot 47 + 25\cdot 47^{2} + 43\cdot 47^{3} + 45\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 18\cdot 47 + 14\cdot 47^{2} + 38\cdot 47^{3} + 14\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 28\cdot 47 + 32\cdot 47^{2} + 8\cdot 47^{3} + 32\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 34\cdot 47 + 21\cdot 47^{2} + 3\cdot 47^{3} + 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 38 + 47 + 25\cdot 47^{2} + 26\cdot 47^{3} + 24\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 38\cdot 47 + 32\cdot 47^{2} + 25\cdot 47^{3} + 6\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,3,5)(4,8,7,6)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,6)(3,7)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,6)(2,7)(3,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,2,3,5)(4,8,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
