Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 3 + 32\cdot 43 + 38\cdot 43^{2} + 16\cdot 43^{3} + 33\cdot 43^{4} + 22\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 4 + 36\cdot 43 + 23\cdot 43^{2} + 24\cdot 43^{3} + 32\cdot 43^{4} + 19\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 28\cdot 43 + 16\cdot 43^{2} + 24\cdot 43^{3} + 14\cdot 43^{4} + 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 13 + 10\cdot 43 + 36\cdot 43^{2} + 22\cdot 43^{3} + 37\cdot 43^{4} +O\left(43^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 31 + 32\cdot 43 + 6\cdot 43^{2} + 20\cdot 43^{3} + 5\cdot 43^{4} + 42\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 14\cdot 43 + 26\cdot 43^{2} + 18\cdot 43^{3} + 28\cdot 43^{4} + 41\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 6\cdot 43 + 19\cdot 43^{2} + 18\cdot 43^{3} + 10\cdot 43^{4} + 23\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 10\cdot 43 + 4\cdot 43^{2} + 26\cdot 43^{3} + 9\cdot 43^{4} + 20\cdot 43^{5} +O\left(43^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,5)(4,6)(7,8)$ |
| $(1,3,2,5)(4,8,6,7)$ |
| $(1,4)(2,6)(3,7)(5,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,2)(3,5)(4,6)(7,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,4)(2,6)(3,7)(5,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,7)(2,8)(3,6)(4,5)$ |
$0$ |
| $2$ |
$4$ |
$(1,3,2,5)(4,8,6,7)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
