Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 11.
Roots:
| $r_{ 1 }$ |
$=$ |
$ 4 + 40\cdot 41 + 37\cdot 41^{3} + 9\cdot 41^{4} + 8\cdot 41^{5} + 4\cdot 41^{6} + 27\cdot 41^{7} + 24\cdot 41^{8} + 13\cdot 41^{9} + 32\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 6 + 14\cdot 41 + 12\cdot 41^{2} + 16\cdot 41^{3} + 33\cdot 41^{4} + 12\cdot 41^{5} + 38\cdot 41^{6} + 34\cdot 41^{7} + 19\cdot 41^{8} + 17\cdot 41^{9} + 15\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 7 + 7\cdot 41 + 7\cdot 41^{2} + 26\cdot 41^{3} + 37\cdot 41^{4} + 39\cdot 41^{5} + 17\cdot 41^{6} + 29\cdot 41^{7} + 38\cdot 41^{8} + 17\cdot 41^{9} + 15\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 19 + 8\cdot 41 + 9\cdot 41^{3} + 31\cdot 41^{4} + 3\cdot 41^{5} + 23\cdot 41^{6} + 23\cdot 41^{7} + 33\cdot 41^{8} + 33\cdot 41^{9} + 8\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 22 + 32\cdot 41 + 40\cdot 41^{2} + 31\cdot 41^{3} + 9\cdot 41^{4} + 37\cdot 41^{5} + 17\cdot 41^{6} + 17\cdot 41^{7} + 7\cdot 41^{8} + 7\cdot 41^{9} + 32\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 34 + 33\cdot 41 + 33\cdot 41^{2} + 14\cdot 41^{3} + 3\cdot 41^{4} + 41^{5} + 23\cdot 41^{6} + 11\cdot 41^{7} + 2\cdot 41^{8} + 23\cdot 41^{9} + 25\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 7 }$ |
$=$ |
$ 35 + 26\cdot 41 + 28\cdot 41^{2} + 24\cdot 41^{3} + 7\cdot 41^{4} + 28\cdot 41^{5} + 2\cdot 41^{6} + 6\cdot 41^{7} + 21\cdot 41^{8} + 23\cdot 41^{9} + 25\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
| $r_{ 8 }$ |
$=$ |
$ 37 + 40\cdot 41^{2} + 3\cdot 41^{3} + 31\cdot 41^{4} + 32\cdot 41^{5} + 36\cdot 41^{6} + 13\cdot 41^{7} + 16\cdot 41^{8} + 27\cdot 41^{9} + 8\cdot 41^{10} +O\left(41^{ 11 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 8 }$
| Cycle notation |
| $(1,5,3,2,8,4,6,7)$ |
| $(1,8)(2,7)(3,6)(4,5)$ |
| $(1,3,8,6)(2,4,7,5)$ |
| $(2,7)(4,5)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 8 }$
| Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $0$ |
| $2$ | $8$ | $(1,2,6,5,8,7,3,4)$ | $0$ |
| $2$ | $8$ | $(1,5,6,7,8,4,3,2)$ | $0$ |
| $2$ | $8$ | $(1,7,3,5,8,2,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
