Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $-1$ |
| Root number: | $-1$ |
| Artin field: | Galois closure of 8.0.12230590464.1 |
| 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
| $f(x)$ | $=$ |
\( x^{8} + 12x^{6} + 36x^{4} + 36x^{2} + 9 \)
|
The roots of $f$ are computed in $\Q_{ 47 }$ to precision 9.
Roots:
| $r_{ 1 }$ | $=$ |
\( 1 + 20\cdot 47 + 35\cdot 47^{2} + 41\cdot 47^{3} + 9\cdot 47^{4} + 42\cdot 47^{5} + 25\cdot 47^{6} + 13\cdot 47^{7} + 32\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 2 }$ | $=$ |
\( 8 + 29\cdot 47 + 9\cdot 47^{2} + 6\cdot 47^{3} + 14\cdot 47^{4} + 5\cdot 47^{5} + 15\cdot 47^{6} + 37\cdot 47^{7} + 42\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 36\cdot 47 + 10\cdot 47^{2} + 29\cdot 47^{3} + 33\cdot 47^{4} + 26\cdot 47^{5} + 20\cdot 47^{6} + 7\cdot 47^{7} + 5\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 4 }$ | $=$ |
\( 22 + 38\cdot 47 + 43\cdot 47^{2} + 35\cdot 47^{3} + 22\cdot 47^{4} + 17\cdot 47^{5} + 5\cdot 47^{6} + 44\cdot 47^{7} + 32\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 5 }$ | $=$ |
\( 25 + 8\cdot 47 + 3\cdot 47^{2} + 11\cdot 47^{3} + 24\cdot 47^{4} + 29\cdot 47^{5} + 41\cdot 47^{6} + 2\cdot 47^{7} + 14\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 6 }$ | $=$ |
\( 35 + 10\cdot 47 + 36\cdot 47^{2} + 17\cdot 47^{3} + 13\cdot 47^{4} + 20\cdot 47^{5} + 26\cdot 47^{6} + 39\cdot 47^{7} + 41\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 7 }$ | $=$ |
\( 39 + 17\cdot 47 + 37\cdot 47^{2} + 40\cdot 47^{3} + 32\cdot 47^{4} + 41\cdot 47^{5} + 31\cdot 47^{6} + 9\cdot 47^{7} + 4\cdot 47^{8} +O(47^{9})\)
|
| $r_{ 8 }$ | $=$ |
\( 46 + 26\cdot 47 + 11\cdot 47^{2} + 5\cdot 47^{3} + 37\cdot 47^{4} + 4\cdot 47^{5} + 21\cdot 47^{6} + 33\cdot 47^{7} + 14\cdot 47^{8} +O(47^{9})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | ✓ |
| $2$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $0$ | |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ | |
| $2$ | $4$ | $(1,5,8,4)(2,3,7,6)$ | $0$ |