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 8.8.29365647704064.1 |
| 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 \(\Q(\sqrt{2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 84x^{6} + 1764x^{4} - 12348x^{2} + 21609 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 66\cdot 73 + 3\cdot 73^{2} + 32\cdot 73^{3} + 60\cdot 73^{4} + 65\cdot 73^{5} + 29\cdot 73^{6} + 10\cdot 73^{7} + 24\cdot 73^{8} + 19\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 48\cdot 73 + 43\cdot 73^{2} + 45\cdot 73^{3} + 46\cdot 73^{4} + 15\cdot 73^{5} + 51\cdot 73^{6} + 50\cdot 73^{7} + 19\cdot 73^{8} + 30\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 20 + 32\cdot 73 + 36\cdot 73^{2} + 44\cdot 73^{3} + 2\cdot 73^{4} + 26\cdot 73^{5} + 52\cdot 73^{6} + 35\cdot 73^{7} + 53\cdot 73^{8} + 71\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 24 + 31\cdot 73 + 53\cdot 73^{2} + 45\cdot 73^{3} + 33\cdot 73^{4} + 34\cdot 73^{5} + 61\cdot 73^{6} + 44\cdot 73^{7} + 73^{8} + 62\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 49 + 41\cdot 73 + 19\cdot 73^{2} + 27\cdot 73^{3} + 39\cdot 73^{4} + 38\cdot 73^{5} + 11\cdot 73^{6} + 28\cdot 73^{7} + 71\cdot 73^{8} + 10\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 53 + 40\cdot 73 + 36\cdot 73^{2} + 28\cdot 73^{3} + 70\cdot 73^{4} + 46\cdot 73^{5} + 20\cdot 73^{6} + 37\cdot 73^{7} + 19\cdot 73^{8} + 73^{9} +O(73^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 24\cdot 73 + 29\cdot 73^{2} + 27\cdot 73^{3} + 26\cdot 73^{4} + 57\cdot 73^{5} + 21\cdot 73^{6} + 22\cdot 73^{7} + 53\cdot 73^{8} + 42\cdot 73^{9} +O(73^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 70 + 6\cdot 73 + 69\cdot 73^{2} + 40\cdot 73^{3} + 12\cdot 73^{4} + 7\cdot 73^{5} + 43\cdot 73^{6} + 62\cdot 73^{7} + 48\cdot 73^{8} + 53\cdot 73^{9} +O(73^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.