Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 41\cdot 47 + 31\cdot 47^{2} + 28\cdot 47^{3} + 38\cdot 47^{4} + 43\cdot 47^{5} + 13\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 5 + 16\cdot 47 + 11\cdot 47^{2} + 27\cdot 47^{3} + 4\cdot 47^{4} + 17\cdot 47^{5} + 25\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 6 + 27\cdot 47 + 4\cdot 47^{2} + 32\cdot 47^{3} + 27\cdot 47^{4} + 14\cdot 47^{5} + 12\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 9 + 5\cdot 47 + 45\cdot 47^{2} + 21\cdot 47^{3} + 35\cdot 47^{4} + 13\cdot 47^{5} + 23\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 11 + 31\cdot 47 + 29\cdot 47^{2} + 15\cdot 47^{3} + 27\cdot 47^{5} + 22\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 29 + 5\cdot 47 + 21\cdot 47^{2} + 22\cdot 47^{3} + 3\cdot 47^{4} + 6\cdot 47^{5} + 32\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 34 + 9\cdot 47 + 46\cdot 47^{2} + 5\cdot 47^{3} + 23\cdot 47^{4} + 18\cdot 47^{5} + 42\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 4\cdot 47 + 45\cdot 47^{2} + 33\cdot 47^{3} + 7\cdot 47^{4} + 16\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,7)(4,8)(5,6)$ |
| $(1,4)(2,8)(3,5)(6,7)$ |
| $(1,3)(2,7)(4,6)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,7)(4,8)(5,6)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,7)(4,6)(5,8)$ | $0$ |
| $2$ | $2$ | $(1,4)(2,8)(3,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,2,6)(3,4,7,8)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
