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 8.8.179068074983424.2 |
| 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
| $f(x)$ | $=$ |
\( x^{8} - 132x^{6} + 4356x^{4} - 47916x^{2} + 131769 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 2\cdot 47 + 25\cdot 47^{2} + 6\cdot 47^{3} + 3\cdot 47^{4} + 40\cdot 47^{5} + 46\cdot 47^{6} + 43\cdot 47^{7} + 39\cdot 47^{8} + 44\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 22\cdot 47 + 25\cdot 47^{2} + 8\cdot 47^{3} + 45\cdot 47^{4} + 6\cdot 47^{5} + 37\cdot 47^{6} + 19\cdot 47^{7} + 5\cdot 47^{8} + 3\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 9 + 45\cdot 47 + 35\cdot 47^{2} + 41\cdot 47^{3} + 30\cdot 47^{4} + 27\cdot 47^{5} + 21\cdot 47^{6} + 5\cdot 47^{7} + 47^{8} + 29\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 18\cdot 47 + 47^{2} + 17\cdot 47^{3} + 47^{4} + 26\cdot 47^{5} + 9\cdot 47^{6} + 32\cdot 47^{7} + 9\cdot 47^{8} + 17\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 28\cdot 47 + 45\cdot 47^{2} + 29\cdot 47^{3} + 45\cdot 47^{4} + 20\cdot 47^{5} + 37\cdot 47^{6} + 14\cdot 47^{7} + 37\cdot 47^{8} + 29\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 38 + 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 16\cdot 47^{4} + 19\cdot 47^{5} + 25\cdot 47^{6} + 41\cdot 47^{7} + 45\cdot 47^{8} + 17\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 41 + 24\cdot 47 + 21\cdot 47^{2} + 38\cdot 47^{3} + 47^{4} + 40\cdot 47^{5} + 9\cdot 47^{6} + 27\cdot 47^{7} + 41\cdot 47^{8} + 43\cdot 47^{9} +O(47^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 + 44\cdot 47 + 21\cdot 47^{2} + 40\cdot 47^{3} + 43\cdot 47^{4} + 6\cdot 47^{5} + 3\cdot 47^{7} + 7\cdot 47^{8} + 2\cdot 47^{9} +O(47^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.