Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 41\cdot 47 + 22\cdot 47^{3} + 39\cdot 47^{4} + 9\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 46\cdot 47 + 9\cdot 47^{2} + 28\cdot 47^{3} + 36\cdot 47^{4} + 46\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 14\cdot 47 + 39\cdot 47^{2} + 10\cdot 47^{3} + 6\cdot 47^{4} + 22\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 17\cdot 47 + 17\cdot 47^{2} + 46\cdot 47^{3} + 2\cdot 47^{4} + 11\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 29\cdot 47 + 29\cdot 47^{2} + 44\cdot 47^{4} + 35\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 32 + 32\cdot 47 + 7\cdot 47^{2} + 36\cdot 47^{3} + 40\cdot 47^{4} + 24\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 43 + 37\cdot 47^{2} + 18\cdot 47^{3} + 10\cdot 47^{4} +O\left(47^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 5\cdot 47 + 46\cdot 47^{2} + 24\cdot 47^{3} + 7\cdot 47^{4} + 37\cdot 47^{5} +O\left(47^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
