Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 7.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 13\cdot 47 + 39\cdot 47^{2} + 32\cdot 47^{3} + 34\cdot 47^{4} + 23\cdot 47^{5} + 33\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 14 + 9\cdot 47 + 4\cdot 47^{2} + 6\cdot 47^{3} + 32\cdot 47^{4} + 39\cdot 47^{5} + 39\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 20 + 29\cdot 47 + 31\cdot 47^{2} + 27\cdot 47^{4} + 47^{5} + 36\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 23 + 31\cdot 47 + 3\cdot 47^{2} + 15\cdot 47^{3} + 44\cdot 47^{4} + 19\cdot 47^{5} + 43\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 15\cdot 47 + 43\cdot 47^{2} + 31\cdot 47^{3} + 2\cdot 47^{4} + 27\cdot 47^{5} + 3\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 17\cdot 47 + 15\cdot 47^{2} + 46\cdot 47^{3} + 19\cdot 47^{4} + 45\cdot 47^{5} + 10\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 33 + 37\cdot 47 + 42\cdot 47^{2} + 40\cdot 47^{3} + 14\cdot 47^{4} + 7\cdot 47^{5} + 7\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 45 + 33\cdot 47 + 7\cdot 47^{2} + 14\cdot 47^{3} + 12\cdot 47^{4} + 23\cdot 47^{5} + 13\cdot 47^{6} +O\left(47^{ 7 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,4)(2,3)(5,8)(6,7)$ |
| $(1,6,8,3)(4,5)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,8)(3,6)$ |
| $(1,3)(2,5)(4,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$4$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-4$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
| $4$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$0$ |
| $4$ |
$4$ |
$(1,5,3,2)(4,6,7,8)$ |
$0$ |
| $4$ |
$4$ |
$(1,2,3,5)(4,8,7,6)$ |
$0$ |
| $4$ |
$4$ |
$(1,4,8,5)(2,6,7,3)$ |
$0$ |
| $4$ |
$4$ |
$(1,6,8,3)(4,5)$ |
$0$ |
| $4$ |
$4$ |
$(1,3,8,6)(4,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
