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 8.0.21667237072994304.1 |
| 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:
| $r_{ 1 }$ | $=$ |
\( 4 + 31\cdot 53 + 36\cdot 53^{2} + 16\cdot 53^{3} + 8\cdot 53^{4} + 43\cdot 53^{5} + 12\cdot 53^{6} + 34\cdot 53^{7} + 25\cdot 53^{8} + 32\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 39\cdot 53 + 47\cdot 53^{2} + 17\cdot 53^{3} + 32\cdot 53^{4} + 34\cdot 53^{5} + 3\cdot 53^{6} + 43\cdot 53^{7} + 52\cdot 53^{8} + 13\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 13 + 7\cdot 53 + 3\cdot 53^{2} + 35\cdot 53^{3} + 46\cdot 53^{4} + 14\cdot 53^{5} + 3\cdot 53^{6} + 22\cdot 53^{7} + 36\cdot 53^{8} + 39\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 20 + 3\cdot 53 + 45\cdot 53^{2} + 40\cdot 53^{3} + 37\cdot 53^{4} + 43\cdot 53^{5} + 48\cdot 53^{6} + 19\cdot 53^{7} + 36\cdot 53^{8} + 34\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 33 + 49\cdot 53 + 7\cdot 53^{2} + 12\cdot 53^{3} + 15\cdot 53^{4} + 9\cdot 53^{5} + 4\cdot 53^{6} + 33\cdot 53^{7} + 16\cdot 53^{8} + 18\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 40 + 45\cdot 53 + 49\cdot 53^{2} + 17\cdot 53^{3} + 6\cdot 53^{4} + 38\cdot 53^{5} + 49\cdot 53^{6} + 30\cdot 53^{7} + 16\cdot 53^{8} + 13\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 48 + 13\cdot 53 + 5\cdot 53^{2} + 35\cdot 53^{3} + 20\cdot 53^{4} + 18\cdot 53^{5} + 49\cdot 53^{6} + 9\cdot 53^{7} + 39\cdot 53^{9} +O(53^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 49 + 21\cdot 53 + 16\cdot 53^{2} + 36\cdot 53^{3} + 44\cdot 53^{4} + 9\cdot 53^{5} + 40\cdot 53^{6} + 18\cdot 53^{7} + 27\cdot 53^{8} + 20\cdot 53^{9} +O(53^{10})\)
|
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 | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |