Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(1280\)\(\medspace = 2^{8} \cdot 5 \) |
| Artin number field: | Galois closure of 8.0.419430400.3 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{5})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 + 4\cdot 41 + 41^{2} + 40\cdot 41^{3} + 31\cdot 41^{4} + 9\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 17\cdot 41 + 35\cdot 41^{2} + 39\cdot 41^{3} + 38\cdot 41^{4} + 27\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 35\cdot 41 + 34\cdot 41^{2} + 32\cdot 41^{3} + 40\cdot 41^{4} + 36\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 29\cdot 41 + 12\cdot 41^{2} + 8\cdot 41^{3} + 23\cdot 41^{4} + 18\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 11\cdot 41 + 28\cdot 41^{2} + 32\cdot 41^{3} + 17\cdot 41^{4} + 22\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 25 + 5\cdot 41 + 6\cdot 41^{2} + 8\cdot 41^{3} + 4\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 26 + 23\cdot 41 + 5\cdot 41^{2} + 41^{3} + 2\cdot 41^{4} + 13\cdot 41^{5} +O(41^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 29 + 36\cdot 41 + 39\cdot 41^{2} + 9\cdot 41^{4} + 31\cdot 41^{5} +O(41^{6})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | $0$ |