Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 26\cdot 43 + 26\cdot 43^{2} + 23\cdot 43^{3} + 24\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 7\cdot 43 + 23\cdot 43^{2} + 27\cdot 43^{3} + 29\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 15 + 8\cdot 43 + 38\cdot 43^{3} + 7\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 37\cdot 43 + 30\cdot 43^{2} + 21\cdot 43^{3} + 15\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 24 + 5\cdot 43 + 12\cdot 43^{2} + 21\cdot 43^{3} + 27\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 28 + 34\cdot 43 + 42\cdot 43^{2} + 4\cdot 43^{3} + 35\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 31 + 35\cdot 43 + 19\cdot 43^{2} + 15\cdot 43^{3} + 13\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 16\cdot 43 + 16\cdot 43^{2} + 19\cdot 43^{3} + 18\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3,8,6)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,3)(4,8)(6,7)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,8,6)(2,5,7,4)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
