Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 25\cdot 43 + 29\cdot 43^{2} + 21\cdot 43^{3} + 33\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 29\cdot 43 + 17\cdot 43^{2} + 10\cdot 43^{3} + 14\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 25 + 42\cdot 43 + 34\cdot 43^{2} + 29\cdot 43^{3} + 34\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 26 + 40\cdot 43 + 27\cdot 43^{2} + 27\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 27 + 20\cdot 43 + 36\cdot 43^{2} + 6\cdot 43^{3} + 14\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 37 + 26\cdot 43 + 4\cdot 43^{2} + 26\cdot 43^{3} + 26\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 40 + 40\cdot 43 + 16\cdot 43^{2} + 25\cdot 43^{3} + 41\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 41 + 31\cdot 43 + 3\cdot 43^{2} + 24\cdot 43^{3} + 3\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2,6,4)(3,8,7,5)$ |
| $(1,3)(2,5)(4,8)(6,7)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,6)(2,4)(3,7)(5,8)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,8)(6,7)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,4)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,6,4)(3,8,7,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
