Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 44\cdot 47 + 27\cdot 47^{2} + 36\cdot 47^{3} + 21\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 6\cdot 47^{3} + 36\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 18 + 25\cdot 47 + 25\cdot 47^{2} + 14\cdot 47^{3} + 44\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 22 + 2\cdot 47 + 30\cdot 47^{2} + 27\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 43\cdot 47 + 34\cdot 47^{2} + 11\cdot 47^{3} + 30\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 3\cdot 47 + 38\cdot 47^{2} + 10\cdot 47^{3} + 16\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 42\cdot 47 + 25\cdot 47^{2} + 7\cdot 47^{3} + 8\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 44 + 25\cdot 47 + 5\cdot 47^{2} + 6\cdot 47^{3} + 4\cdot 47^{4} +O\left(47^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,7,8,5)(2,4,3,6)$ |
| $(1,3,8,2)(4,5,6,7)$ |
| $(1,8)(2,3)(4,6)(5,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,3)(4,6)(5,7)$ | $-2$ |
| $2$ | $4$ | $(1,3,8,2)(4,5,6,7)$ | $0$ |
| $2$ | $4$ | $(1,7,8,5)(2,4,3,6)$ | $0$ |
| $2$ | $4$ | $(1,6,8,4)(2,7,3,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
