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 8.8.29721861554176.1 |
| 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 $\Q_{ 79 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 32\cdot 79 + 64\cdot 79^{2} + 77\cdot 79^{3} + 43\cdot 79^{4} + 47\cdot 79^{6} + 15\cdot 79^{7} + 32\cdot 79^{8} + 40\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 14 + 46\cdot 79 + 52\cdot 79^{2} + 44\cdot 79^{3} + 53\cdot 79^{4} + 15\cdot 79^{5} + 20\cdot 79^{6} + 65\cdot 79^{7} + 62\cdot 79^{8} + 51\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 16 + 54\cdot 79 + 64\cdot 79^{2} + 18\cdot 79^{3} + 27\cdot 79^{4} + 18\cdot 79^{5} + 43\cdot 79^{6} + 29\cdot 79^{7} + 61\cdot 79^{8} + 64\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 61\cdot 79 + 76\cdot 79^{2} + 61\cdot 79^{3} + 45\cdot 79^{4} + 75\cdot 79^{5} + 37\cdot 79^{6} + 66\cdot 79^{7} + 45\cdot 79^{8} + 58\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 46 + 17\cdot 79 + 2\cdot 79^{2} + 17\cdot 79^{3} + 33\cdot 79^{4} + 3\cdot 79^{5} + 41\cdot 79^{6} + 12\cdot 79^{7} + 33\cdot 79^{8} + 20\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 63 + 24\cdot 79 + 14\cdot 79^{2} + 60\cdot 79^{3} + 51\cdot 79^{4} + 60\cdot 79^{5} + 35\cdot 79^{6} + 49\cdot 79^{7} + 17\cdot 79^{8} + 14\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 65 + 32\cdot 79 + 26\cdot 79^{2} + 34\cdot 79^{3} + 25\cdot 79^{4} + 63\cdot 79^{5} + 58\cdot 79^{6} + 13\cdot 79^{7} + 16\cdot 79^{8} + 27\cdot 79^{9} +O(79^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 46\cdot 79 + 14\cdot 79^{2} + 79^{3} + 35\cdot 79^{4} + 78\cdot 79^{5} + 31\cdot 79^{6} + 63\cdot 79^{7} + 46\cdot 79^{8} + 38\cdot 79^{9} +O(79^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | |
| $2$ | $4$ | $(1,4,8,5)(2,3,7,6)$ | $0$ | |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |