Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(112896\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 7^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.359729184374784.1 |
| 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 \(\Q(\sqrt{6}, \sqrt{14})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 84x^{6} + 1890x^{4} + 10584x^{2} + 1764 \)
|
The roots of $f$ are computed in $\Q_{ 43 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 29\cdot 43 + 2\cdot 43^{2} + 5\cdot 43^{3} + 34\cdot 43^{4} + 20\cdot 43^{5} + 26\cdot 43^{6} + 31\cdot 43^{7} + 39\cdot 43^{8} + 4\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 3\cdot 43 + 43^{2} + 39\cdot 43^{3} + 21\cdot 43^{4} + 28\cdot 43^{5} + 43^{6} + 10\cdot 43^{7} + 14\cdot 43^{8} + 3\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 14\cdot 43 + 24\cdot 43^{2} + 30\cdot 43^{3} + 6\cdot 43^{4} + 17\cdot 43^{5} + 18\cdot 43^{6} + 25\cdot 43^{7} + 32\cdot 43^{8} + 6\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 36\cdot 43 + 28\cdot 43^{3} + 43^{4} + 19\cdot 43^{5} + 8\cdot 43^{6} + 11\cdot 43^{7} + 9\cdot 43^{8} + 18\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 6\cdot 43 + 42\cdot 43^{2} + 14\cdot 43^{3} + 41\cdot 43^{4} + 23\cdot 43^{5} + 34\cdot 43^{6} + 31\cdot 43^{7} + 33\cdot 43^{8} + 24\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 31 + 28\cdot 43 + 18\cdot 43^{2} + 12\cdot 43^{3} + 36\cdot 43^{4} + 25\cdot 43^{5} + 24\cdot 43^{6} + 17\cdot 43^{7} + 10\cdot 43^{8} + 36\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 32 + 39\cdot 43 + 41\cdot 43^{2} + 3\cdot 43^{3} + 21\cdot 43^{4} + 14\cdot 43^{5} + 41\cdot 43^{6} + 32\cdot 43^{7} + 28\cdot 43^{8} + 39\cdot 43^{9} +O(43^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 38 + 13\cdot 43 + 40\cdot 43^{2} + 37\cdot 43^{3} + 8\cdot 43^{4} + 22\cdot 43^{5} + 16\cdot 43^{6} + 11\cdot 43^{7} + 3\cdot 43^{8} + 38\cdot 43^{9} +O(43^{10})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |