Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 9.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 8 + 28\cdot 41 + 19\cdot 41^{2} + 40\cdot 41^{3} + 5\cdot 41^{4} + 26\cdot 41^{5} + 27\cdot 41^{6} + 15\cdot 41^{7} + 21\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 9 + 34\cdot 41 + 37\cdot 41^{2} + 41^{3} + 24\cdot 41^{4} + 4\cdot 41^{5} + 16\cdot 41^{6} + 20\cdot 41^{7} + 26\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 17 + 31\cdot 41 + 28\cdot 41^{2} + 16\cdot 41^{3} + 11\cdot 41^{4} + 4\cdot 41^{5} + 8\cdot 41^{6} + 25\cdot 41^{7} + 17\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 18 + 32\cdot 41^{2} + 21\cdot 41^{3} + 23\cdot 41^{4} + 18\cdot 41^{5} + 33\cdot 41^{6} + 18\cdot 41^{7} + 36\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 23 + 40\cdot 41 + 8\cdot 41^{2} + 19\cdot 41^{3} + 17\cdot 41^{4} + 22\cdot 41^{5} + 7\cdot 41^{6} + 22\cdot 41^{7} + 4\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 24 + 9\cdot 41 + 12\cdot 41^{2} + 24\cdot 41^{3} + 29\cdot 41^{4} + 36\cdot 41^{5} + 32\cdot 41^{6} + 15\cdot 41^{7} + 23\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 32 + 6\cdot 41 + 3\cdot 41^{2} + 39\cdot 41^{3} + 16\cdot 41^{4} + 36\cdot 41^{5} + 24\cdot 41^{6} + 20\cdot 41^{7} + 14\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 33 + 12\cdot 41 + 21\cdot 41^{2} + 35\cdot 41^{4} + 14\cdot 41^{5} + 13\cdot 41^{6} + 25\cdot 41^{7} + 19\cdot 41^{8} +O\left(41^{ 9 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,2,6,5,8,7,3,4)$ |
| $(2,7)(4,5)$ |
| $(1,6,8,3)(2,5,7,4)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character values |
| | |
$c1$ |
$c2$ |
| $1$ |
$1$ |
$()$ |
$2$ |
$2$ |
| $1$ |
$2$ |
$(1,8)(2,7)(3,6)(4,5)$ |
$-2$ |
$-2$ |
| $2$ |
$2$ |
$(2,7)(4,5)$ |
$0$ |
$0$ |
| $1$ |
$4$ |
$(1,6,8,3)(2,5,7,4)$ |
$2 \zeta_{4}$ |
$-2 \zeta_{4}$ |
| $1$ |
$4$ |
$(1,3,8,6)(2,4,7,5)$ |
$-2 \zeta_{4}$ |
$2 \zeta_{4}$ |
| $2$ |
$4$ |
$(1,6,8,3)(2,4,7,5)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,2,6,5,8,7,3,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,3,2,8,4,6,7)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,7,3,5,8,2,6,4)$ |
$0$ |
$0$ |
| $2$ |
$8$ |
$(1,5,6,7,8,4,3,2)$ |
$0$ |
$0$ |
The blue line marks the conjugacy class containing complex conjugation.
