Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(3200\)\(\medspace = 2^{7} \cdot 5^{2} \) |
| Artin stem field: | Galois closure of 8.0.4096000000.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.40.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\zeta_{8})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 10x^{6} + 50x^{4} - 100x^{2} + 100 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 15\cdot 41 + 32\cdot 41^{2} + 2\cdot 41^{3} + 8\cdot 41^{4} + 32\cdot 41^{5} + 2\cdot 41^{6} + 39\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 4\cdot 41 + 2\cdot 41^{2} + 2\cdot 41^{3} + 9\cdot 41^{4} + 33\cdot 41^{5} + 22\cdot 41^{6} + 9\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 + 25\cdot 41 + 40\cdot 41^{2} + 19\cdot 41^{3} + 30\cdot 41^{4} + 19\cdot 41^{5} + 30\cdot 41^{6} + 14\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 34\cdot 41 + 15\cdot 41^{3} + 20\cdot 41^{4} + 29\cdot 41^{5} + 8\cdot 41^{6} +O(41^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 6\cdot 41 + 40\cdot 41^{2} + 25\cdot 41^{3} + 20\cdot 41^{4} + 11\cdot 41^{5} + 32\cdot 41^{6} + 40\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 15\cdot 41 + 21\cdot 41^{3} + 10\cdot 41^{4} + 21\cdot 41^{5} + 10\cdot 41^{6} + 26\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 34 + 36\cdot 41 + 38\cdot 41^{2} + 38\cdot 41^{3} + 31\cdot 41^{4} + 7\cdot 41^{5} + 18\cdot 41^{6} + 31\cdot 41^{7} +O(41^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 40 + 25\cdot 41 + 8\cdot 41^{2} + 38\cdot 41^{3} + 32\cdot 41^{4} + 8\cdot 41^{5} + 38\cdot 41^{6} + 41^{7} +O(41^{8})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,4)(5,7)(6,8)$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.