Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 6 + 16\cdot 41 + 3\cdot 41^{2} + 3\cdot 41^{3} + 4\cdot 41^{4} + 3\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 12 + 34\cdot 41 + 17\cdot 41^{2} + 2\cdot 41^{3} + 16\cdot 41^{4} + 36\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 16 + 23\cdot 41 + 20\cdot 41^{2} + 12\cdot 41^{3} + 37\cdot 41^{4} + 28\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 14\cdot 41 + 40\cdot 41^{2} + 35\cdot 41^{3} + 8\cdot 41^{4} + 16\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 26\cdot 41 + 5\cdot 41^{3} + 32\cdot 41^{4} + 24\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 25 + 17\cdot 41 + 20\cdot 41^{2} + 28\cdot 41^{3} + 3\cdot 41^{4} + 12\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 29 + 6\cdot 41 + 23\cdot 41^{2} + 38\cdot 41^{3} + 24\cdot 41^{4} + 4\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 35 + 24\cdot 41 + 37\cdot 41^{2} + 37\cdot 41^{3} + 36\cdot 41^{4} + 37\cdot 41^{5} +O\left(41^{ 6 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,2)(3,6)(4,5)(7,8)$ |
| $(1,3)(2,4)(5,7)(6,8)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
| $1$ |
$1$ |
$()$ |
$2$ |
| $1$ |
$2$ |
$(1,7)(2,8)(3,5)(4,6)$ |
$-2$ |
| $2$ |
$2$ |
$(1,2)(3,6)(4,5)(7,8)$ |
$0$ |
| $2$ |
$2$ |
$(1,3)(2,4)(5,7)(6,8)$ |
$0$ |
| $2$ |
$4$ |
$(1,4,7,6)(2,3,8,5)$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
