Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(57600\)\(\medspace = 2^{8} \cdot 3^{2} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $1$ |
| Artin field: | Galois closure of 8.8.47775744000000.3 |
| 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 \(\Q(\sqrt{5}, \sqrt{6})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 60x^{6} + 1170x^{4} - 9000x^{2} + 22500 \)
|
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 6\cdot 19 + 14\cdot 19^{2} + 19^{3} + 12\cdot 19^{4} + 13\cdot 19^{5} + 18\cdot 19^{6} + 9\cdot 19^{8} + 8\cdot 19^{9} +O(19^{10})\)
| $r_{ 2 }$ |
$=$ |
\( 4 + 7\cdot 19 + 3\cdot 19^{2} + 15\cdot 19^{3} + 19^{4} + 10\cdot 19^{5} + 13\cdot 19^{6} + 17\cdot 19^{7} + 10\cdot 19^{8} + 11\cdot 19^{9} +O(19^{10})\)
| $r_{ 3 }$ |
$=$ |
\( 6 + 12\cdot 19 + 19^{2} + 18\cdot 19^{3} + 13\cdot 19^{4} + 13\cdot 19^{5} + 4\cdot 19^{6} + 10\cdot 19^{7} + 2\cdot 19^{9} +O(19^{10})\)
| $r_{ 4 }$ |
$=$ |
\( 8 + 2\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 5\cdot 19^{4} + 10\cdot 19^{5} + 9\cdot 19^{7} + 17\cdot 19^{8} + 12\cdot 19^{9} +O(19^{10})\)
| $r_{ 5 }$ |
$=$ |
\( 11 + 16\cdot 19 + 16\cdot 19^{2} + 15\cdot 19^{3} + 13\cdot 19^{4} + 8\cdot 19^{5} + 18\cdot 19^{6} + 9\cdot 19^{7} + 19^{8} + 6\cdot 19^{9} +O(19^{10})\)
| $r_{ 6 }$ |
$=$ |
\( 13 + 6\cdot 19 + 17\cdot 19^{2} + 5\cdot 19^{4} + 5\cdot 19^{5} + 14\cdot 19^{6} + 8\cdot 19^{7} + 18\cdot 19^{8} + 16\cdot 19^{9} +O(19^{10})\)
| $r_{ 7 }$ |
$=$ |
\( 15 + 11\cdot 19 + 15\cdot 19^{2} + 3\cdot 19^{3} + 17\cdot 19^{4} + 8\cdot 19^{5} + 5\cdot 19^{6} + 19^{7} + 8\cdot 19^{8} + 7\cdot 19^{9} +O(19^{10})\)
| $r_{ 8 }$ |
$=$ |
\( 18 + 12\cdot 19 + 4\cdot 19^{2} + 17\cdot 19^{3} + 6\cdot 19^{4} + 5\cdot 19^{5} + 18\cdot 19^{7} + 9\cdot 19^{8} + 10\cdot 19^{9} +O(19^{10})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.