Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(278784\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.21667237072994304.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $Q_8$ |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{6}, \sqrt{11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 132x^{6} + 4356x^{4} - 30492x^{2} + 27225 \)
|
The roots of $f$ are computed in $\Q_{ 53 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 43\cdot 53 + 21\cdot 53^{2} + 6\cdot 53^{3} + 21\cdot 53^{4} + 46\cdot 53^{5} + 40\cdot 53^{6} + 4\cdot 53^{7} + 52\cdot 53^{8} + 50\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 44\cdot 53 + 31\cdot 53^{2} + 37\cdot 53^{3} + 30\cdot 53^{4} + 5\cdot 53^{5} + 14\cdot 53^{6} + 47\cdot 53^{7} + 42\cdot 53^{8} + 8\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 17 + 7\cdot 53 + 34\cdot 53^{2} + 40\cdot 53^{3} + 15\cdot 53^{4} + 24\cdot 53^{6} + 4\cdot 53^{7} + 45\cdot 53^{8} + 33\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 34\cdot 53 + 45\cdot 53^{2} + 31\cdot 53^{3} + 10\cdot 53^{4} + 6\cdot 53^{5} + 18\cdot 53^{6} + 34\cdot 53^{7} + 32\cdot 53^{8} + 11\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 34 + 18\cdot 53 + 7\cdot 53^{2} + 21\cdot 53^{3} + 42\cdot 53^{4} + 46\cdot 53^{5} + 34\cdot 53^{6} + 18\cdot 53^{7} + 20\cdot 53^{8} + 41\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 45\cdot 53 + 18\cdot 53^{2} + 12\cdot 53^{3} + 37\cdot 53^{4} + 52\cdot 53^{5} + 28\cdot 53^{6} + 48\cdot 53^{7} + 7\cdot 53^{8} + 19\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 8\cdot 53 + 21\cdot 53^{2} + 15\cdot 53^{3} + 22\cdot 53^{4} + 47\cdot 53^{5} + 38\cdot 53^{6} + 5\cdot 53^{7} + 10\cdot 53^{8} + 44\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 44 + 9\cdot 53 + 31\cdot 53^{2} + 46\cdot 53^{3} + 31\cdot 53^{4} + 6\cdot 53^{5} + 12\cdot 53^{6} + 48\cdot 53^{7} + 2\cdot 53^{9} +O(53^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.