Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 5.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 9 + 30\cdot 41 + 24\cdot 41^{2} + 12\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 13 + 10\cdot 41 + 13\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 14 + 35\cdot 41 + 8\cdot 41^{2} + 35\cdot 41^{3} + 10\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 16 + 22\cdot 41 + 39\cdot 41^{2} + 22\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 25 + 18\cdot 41 + 41^{2} + 18\cdot 41^{3} + 39\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 27 + 5\cdot 41 + 32\cdot 41^{2} + 5\cdot 41^{3} + 30\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 28 + 30\cdot 41 + 40\cdot 41^{2} + 27\cdot 41^{3} + 20\cdot 41^{4} +O\left(41^{ 5 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 32 + 10\cdot 41 + 16\cdot 41^{2} + 28\cdot 41^{3} + 41^{4} +O\left(41^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,3)(2,5)(4,7)(6,8)$ |
| $(1,2,8,7)(3,4,6,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
| $2$ |
$2$ |
$(1,3)(2,5)(4,7)(6,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,5)(2,6)(3,7)(4,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
