Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 2 + 13\cdot 43 + 36\cdot 43^{2} + 24\cdot 43^{3} + 38\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 7 + 3\cdot 43 + 9\cdot 43^{2} + 10\cdot 43^{3} + 25\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 13 + 33\cdot 43 + 33\cdot 43^{2} + 15\cdot 43^{3} + 10\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 23\cdot 43 + 6\cdot 43^{2} + 43^{3} + 40\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 26 + 19\cdot 43 + 36\cdot 43^{2} + 41\cdot 43^{3} + 2\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 9\cdot 43 + 9\cdot 43^{2} + 27\cdot 43^{3} + 32\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 37 + 39\cdot 43 + 33\cdot 43^{2} + 32\cdot 43^{3} + 17\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 42 + 29\cdot 43 + 6\cdot 43^{2} + 18\cdot 43^{3} + 4\cdot 43^{4} +O\left(43^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,4)(5,6)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
| $(1,5)(2,7)(3,6)(4,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,4)(2,3)(5,8)(6,7)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,4)(5,6)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,7)(3,6)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,7,4,6)(2,5,3,8)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
