Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 1 + 21\cdot 41 + 29\cdot 41^{2} + 2\cdot 41^{3} + 12\cdot 41^{4} + 16\cdot 41^{5} + 41^{6} + 25\cdot 41^{7} + 9\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 2 + 39\cdot 41 + 38\cdot 41^{2} + 6\cdot 41^{3} + 30\cdot 41^{4} + 19\cdot 41^{5} + 8\cdot 41^{6} + 32\cdot 41^{7} + 21\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 10 + 2\cdot 41 + 35\cdot 41^{2} + 4\cdot 41^{3} + 27\cdot 41^{4} + 16\cdot 41^{5} + 5\cdot 41^{6} + 10\cdot 41^{7} + 28\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 11 + 15\cdot 41 + 28\cdot 41^{2} + 34\cdot 41^{3} + 6\cdot 41^{4} + 24\cdot 41^{5} + 40\cdot 41^{6} + 38\cdot 41^{7} + 14\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 30 + 25\cdot 41 + 12\cdot 41^{2} + 6\cdot 41^{3} + 34\cdot 41^{4} + 16\cdot 41^{5} + 2\cdot 41^{7} + 26\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 31 + 38\cdot 41 + 5\cdot 41^{2} + 36\cdot 41^{3} + 13\cdot 41^{4} + 24\cdot 41^{5} + 35\cdot 41^{6} + 30\cdot 41^{7} + 12\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 39 + 41 + 2\cdot 41^{2} + 34\cdot 41^{3} + 10\cdot 41^{4} + 21\cdot 41^{5} + 32\cdot 41^{6} + 8\cdot 41^{7} + 19\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 40 + 19\cdot 41 + 11\cdot 41^{2} + 38\cdot 41^{3} + 28\cdot 41^{4} + 24\cdot 41^{5} + 39\cdot 41^{6} + 15\cdot 41^{7} + 31\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,7,8,2)(3,5,6,4)$ |
| $(3,6)(4,5)$ |
| $(1,5,7,6,8,4,2,3)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(3,6)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,7,8,2)(3,5,6,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,2,8,7)(3,4,6,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,7,8,2)(3,4,6,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,7,6,8,4,2,3)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,6,2,5,8,3,7,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,2,3,8,4,7,6)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,3,7,5,8,6,2,4)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
