Basic invariants
| Dimension: | $2$ |
| Group: | $C_8:C_2$ |
| Conductor: | \(6400\)\(\medspace = 2^{8} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.327680000000.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $C_8:C_2$ |
| Parity: | even |
| Determinant: | 1.20.4t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{5})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 20x^{6} + 130x^{4} + 280x^{2} + 20 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 21\cdot 41 + 29\cdot 41^{2} + 2\cdot 41^{3} + 12\cdot 41^{4} + 16\cdot 41^{5} + 41^{6} + 25\cdot 41^{7} + 9\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 + 39\cdot 41 + 38\cdot 41^{2} + 6\cdot 41^{3} + 30\cdot 41^{4} + 19\cdot 41^{5} + 8\cdot 41^{6} + 32\cdot 41^{7} + 21\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 2\cdot 41 + 35\cdot 41^{2} + 4\cdot 41^{3} + 27\cdot 41^{4} + 16\cdot 41^{5} + 5\cdot 41^{6} + 10\cdot 41^{7} + 28\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 11 + 15\cdot 41 + 28\cdot 41^{2} + 34\cdot 41^{3} + 6\cdot 41^{4} + 24\cdot 41^{5} + 40\cdot 41^{6} + 38\cdot 41^{7} + 14\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 30 + 25\cdot 41 + 12\cdot 41^{2} + 6\cdot 41^{3} + 34\cdot 41^{4} + 16\cdot 41^{5} + 2\cdot 41^{7} + 26\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 38\cdot 41 + 5\cdot 41^{2} + 36\cdot 41^{3} + 13\cdot 41^{4} + 24\cdot 41^{5} + 35\cdot 41^{6} + 30\cdot 41^{7} + 12\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 41 + 2\cdot 41^{2} + 34\cdot 41^{3} + 10\cdot 41^{4} + 21\cdot 41^{5} + 32\cdot 41^{6} + 8\cdot 41^{7} + 19\cdot 41^{8} +O(41^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 40 + 19\cdot 41 + 11\cdot 41^{2} + 38\cdot 41^{3} + 28\cdot 41^{4} + 24\cdot 41^{5} + 39\cdot 41^{6} + 15\cdot 41^{7} + 31\cdot 41^{8} +O(41^{9})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 8 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 8 }$ | Character value |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,5,7,6,8,4,2,3)$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $0$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $0$ |
| $2$ | $8$ | $(1,3,7,5,8,6,2,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.