Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(30976\)\(\medspace = 2^{8} \cdot 11^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.29721861554176.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{11})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 44x^{6} + 308x^{4} + 484x^{2} + 121 \)
|
The roots of $f$ are computed in $\Q_{ 89 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 67\cdot 89 + 36\cdot 89^{2} + 87\cdot 89^{3} + 46\cdot 89^{4} + 33\cdot 89^{5} + 65\cdot 89^{6} + 57\cdot 89^{7} + 16\cdot 89^{8} + 46\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 5 + 75\cdot 89 + 27\cdot 89^{2} + 73\cdot 89^{3} + 7\cdot 89^{4} + 11\cdot 89^{5} + 5\cdot 89^{6} + 3\cdot 89^{7} + 29\cdot 89^{8} + 22\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 31 + 47\cdot 89 + 8\cdot 89^{2} + 54\cdot 89^{3} + 87\cdot 89^{4} + 88\cdot 89^{5} + 20\cdot 89^{6} + 3\cdot 89^{7} + 82\cdot 89^{8} + 69\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 37 + 80\cdot 89 + 5\cdot 89^{2} + 44\cdot 89^{3} + 10\cdot 89^{4} + 46\cdot 89^{5} + 46\cdot 89^{6} + 81\cdot 89^{7} + 52\cdot 89^{8} + 18\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 52 + 8\cdot 89 + 83\cdot 89^{2} + 44\cdot 89^{3} + 78\cdot 89^{4} + 42\cdot 89^{5} + 42\cdot 89^{6} + 7\cdot 89^{7} + 36\cdot 89^{8} + 70\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 58 + 41\cdot 89 + 80\cdot 89^{2} + 34\cdot 89^{3} + 89^{4} + 68\cdot 89^{6} + 85\cdot 89^{7} + 6\cdot 89^{8} + 19\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 84 + 13\cdot 89 + 61\cdot 89^{2} + 15\cdot 89^{3} + 81\cdot 89^{4} + 77\cdot 89^{5} + 83\cdot 89^{6} + 85\cdot 89^{7} + 59\cdot 89^{8} + 66\cdot 89^{9} +O(89^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 87 + 21\cdot 89 + 52\cdot 89^{2} + 89^{3} + 42\cdot 89^{4} + 55\cdot 89^{5} + 23\cdot 89^{6} + 31\cdot 89^{7} + 72\cdot 89^{8} + 42\cdot 89^{9} +O(89^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |