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 8.0.5416809268248576.2 |
| 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 \(\Q(\sqrt{2}, \sqrt{33})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 132x^{6} + 5346x^{4} + 69696x^{2} + 278784 \)
|
The roots of $f$ are computed in $\Q_{ 41 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 8 + 30\cdot 41 + 7\cdot 41^{2} + 10\cdot 41^{3} + 37\cdot 41^{4} + 21\cdot 41^{5} + 29\cdot 41^{6} + 4\cdot 41^{7} + 28\cdot 41^{8} + 37\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 15\cdot 41 + 22\cdot 41^{2} + 38\cdot 41^{3} + 36\cdot 41^{4} + 7\cdot 41^{5} + 35\cdot 41^{6} + 32\cdot 41^{7} + 41^{8} + 28\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 9\cdot 41 + 13\cdot 41^{2} + 32\cdot 41^{3} + 4\cdot 41^{4} + 13\cdot 41^{5} + 7\cdot 41^{6} + 22\cdot 41^{7} + 30\cdot 41^{8} + 13\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 22\cdot 41 + 21\cdot 41^{2} + 28\cdot 41^{3} + 25\cdot 41^{4} + 6\cdot 41^{5} + 41^{6} + 6\cdot 41^{7} + 10\cdot 41^{8} + 24\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 21 + 18\cdot 41 + 19\cdot 41^{2} + 12\cdot 41^{3} + 15\cdot 41^{4} + 34\cdot 41^{5} + 39\cdot 41^{6} + 34\cdot 41^{7} + 30\cdot 41^{8} + 16\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 29 + 31\cdot 41 + 27\cdot 41^{2} + 8\cdot 41^{3} + 36\cdot 41^{4} + 27\cdot 41^{5} + 33\cdot 41^{6} + 18\cdot 41^{7} + 10\cdot 41^{8} + 27\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 30 + 25\cdot 41 + 18\cdot 41^{2} + 2\cdot 41^{3} + 4\cdot 41^{4} + 33\cdot 41^{5} + 5\cdot 41^{6} + 8\cdot 41^{7} + 39\cdot 41^{8} + 12\cdot 41^{9} +O(41^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 33 + 10\cdot 41 + 33\cdot 41^{2} + 30\cdot 41^{3} + 3\cdot 41^{4} + 19\cdot 41^{5} + 11\cdot 41^{6} + 36\cdot 41^{7} + 12\cdot 41^{8} + 3\cdot 41^{9} +O(41^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |