Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(288\)\(\medspace = 2^{5} \cdot 3^{2} \) |
| Artin stem field: | Galois closure of 8.2.143327232.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.432.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 6x^{4} - 3 \)
|
The roots of $f$ are computed in $\Q_{ 61 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 13 + 45\cdot 61 + 2\cdot 61^{2} + 15\cdot 61^{3} + 47\cdot 61^{4} + 13\cdot 61^{5} + 54\cdot 61^{6} + 24\cdot 61^{7} + 37\cdot 61^{8} + 53\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 21 + 30\cdot 61 + 40\cdot 61^{2} + 48\cdot 61^{3} + 29\cdot 61^{4} + 34\cdot 61^{5} + 59\cdot 61^{6} + 56\cdot 61^{7} + 58\cdot 61^{8} + 2\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 25 + 21\cdot 61 + 32\cdot 61^{2} + 40\cdot 61^{3} + 19\cdot 61^{4} + 21\cdot 61^{5} + 41\cdot 61^{6} + 29\cdot 61^{7} + 34\cdot 61^{8} + 54\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 30 + 38\cdot 61 + 2\cdot 61^{2} + 18\cdot 61^{3} + 39\cdot 61^{4} + 2\cdot 61^{5} + 40\cdot 61^{6} + 20\cdot 61^{7} + 9\cdot 61^{8} + 46\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 31 + 22\cdot 61 + 58\cdot 61^{2} + 42\cdot 61^{3} + 21\cdot 61^{4} + 58\cdot 61^{5} + 20\cdot 61^{6} + 40\cdot 61^{7} + 51\cdot 61^{8} + 14\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 36 + 39\cdot 61 + 28\cdot 61^{2} + 20\cdot 61^{3} + 41\cdot 61^{4} + 39\cdot 61^{5} + 19\cdot 61^{6} + 31\cdot 61^{7} + 26\cdot 61^{8} + 6\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 40 + 30\cdot 61 + 20\cdot 61^{2} + 12\cdot 61^{3} + 31\cdot 61^{4} + 26\cdot 61^{5} + 61^{6} + 4\cdot 61^{7} + 2\cdot 61^{8} + 58\cdot 61^{9} +O(61^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 48 + 15\cdot 61 + 58\cdot 61^{2} + 45\cdot 61^{3} + 13\cdot 61^{4} + 47\cdot 61^{5} + 6\cdot 61^{6} + 36\cdot 61^{7} + 23\cdot 61^{8} + 7\cdot 61^{9} +O(61^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,4,8,5)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,5,7,3,8,4,2,6)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,4,7,6,8,5,2,3)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.