Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 34\cdot 47 + 33\cdot 47^{3} + 8\cdot 47^{4} + 6\cdot 47^{5} + 23\cdot 47^{6} + 28\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 37\cdot 47 + 38\cdot 47^{2} + 20\cdot 47^{4} + 38\cdot 47^{5} + 34\cdot 47^{6} + 12\cdot 47^{7} + 12\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 5\cdot 47 + 42\cdot 47^{2} + 31\cdot 47^{3} + 15\cdot 47^{4} + 6\cdot 47^{5} + 3\cdot 47^{7} + 44\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 12\cdot 47 + 45\cdot 47^{2} + 45\cdot 47^{3} + 32\cdot 47^{4} + 19\cdot 47^{5} + 17\cdot 47^{6} + 5\cdot 47^{7} + 46\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 4\cdot 47 + 13\cdot 47^{2} + 16\cdot 47^{3} + 37\cdot 47^{4} + 29\cdot 47^{5} + 20\cdot 47^{6} + 42\cdot 47^{7} + 46\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 35 + 37\cdot 47 + 2\cdot 47^{2} + 46\cdot 47^{3} + 12\cdot 47^{4} + 35\cdot 47^{5} + 21\cdot 47^{6} + 10\cdot 47^{7} + 19\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 42 + 4\cdot 47 + 22\cdot 47^{2} + 4\cdot 47^{3} + 15\cdot 47^{4} + 18\cdot 47^{5} + 7\cdot 47^{6} + 24\cdot 47^{7} + 4\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 4\cdot 47 + 23\cdot 47^{2} + 9\cdot 47^{3} + 45\cdot 47^{4} + 33\cdot 47^{5} + 15\cdot 47^{6} + 42\cdot 47^{7} + 33\cdot 47^{8} +O\left(47^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8,5,7)(2,6,3,4)$ |
| $(1,5)(2,3)(4,6)(7,8)$ |
| $(2,3)(4,6)$ |
| $(1,6,8,3,5,4,7,2)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,5)(2,3)(4,6)(7,8)$ | $-2$ |
| $2$ | $2$ | $(2,3)(4,6)$ | $0$ |
| $1$ | $4$ | $(1,8,5,7)(2,6,3,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,5,8)(2,4,3,6)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,8,5,7)(2,4,3,6)$ | $0$ |
| $2$ | $8$ | $(1,6,8,3,5,4,7,2)$ | $0$ |
| $2$ | $8$ | $(1,3,7,6,5,2,8,4)$ | $0$ |
| $2$ | $8$ | $(1,6,7,2,5,4,8,3)$ | $0$ |
| $2$ | $8$ | $(1,2,8,6,5,3,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
