Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2112\)\(\medspace = 2^{6} \cdot 3 \cdot 11 \) |
| Artin number field: | Galois closure of 8.0.8635613184.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{-11})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 2\cdot 53 + 12\cdot 53^{2} + 2\cdot 53^{3} + 33\cdot 53^{4} + 28\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 18 + 24\cdot 53 + 39\cdot 53^{2} + 36\cdot 53^{3} + 32\cdot 53^{4} + 43\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 23\cdot 53 + 48\cdot 53^{2} + 3\cdot 53^{3} + 28\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 21 + 19\cdot 53 + 50\cdot 53^{2} + 33\cdot 53^{3} + 42\cdot 53^{4} + 14\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 32 + 33\cdot 53 + 2\cdot 53^{2} + 19\cdot 53^{3} + 10\cdot 53^{4} + 38\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 33 + 29\cdot 53 + 4\cdot 53^{2} + 49\cdot 53^{3} + 52\cdot 53^{4} + 24\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 35 + 28\cdot 53 + 13\cdot 53^{2} + 16\cdot 53^{3} + 20\cdot 53^{4} + 9\cdot 53^{5} +O(53^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 42 + 50\cdot 53 + 40\cdot 53^{2} + 50\cdot 53^{3} + 19\cdot 53^{4} + 24\cdot 53^{5} +O(53^{6})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,6)(2,5)(3,8)(4,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,8)(4,5)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,2)(3,5)(4,6)(7,8)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |