Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Artin stem field: | Galois closure of 8.2.36691771392.3 |
| 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 4.0.1728.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 12x^{4} - 12 \)
|
The roots of $f$ are computed in $\Q_{ 13 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 3\cdot 13 + 6\cdot 13^{2} + 4\cdot 13^{3} + 11\cdot 13^{4} + 10\cdot 13^{5} + 13^{6} + 3\cdot 13^{7} + 5\cdot 13^{8} + 7\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 3\cdot 13^{2} + 8\cdot 13^{3} + 8\cdot 13^{4} + 13^{5} + 6\cdot 13^{6} + 11\cdot 13^{7} + 11\cdot 13^{8} + 6\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 4 + 3\cdot 13 + 10\cdot 13^{2} + 13^{3} + 7\cdot 13^{4} + 6\cdot 13^{5} + 8\cdot 13^{6} + 10\cdot 13^{7} + 9\cdot 13^{8} + 4\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 6 + 2\cdot 13 + 6\cdot 13^{2} + 13^{3} + 3\cdot 13^{4} + 10\cdot 13^{5} + 13^{6} + 3\cdot 13^{7} + 12\cdot 13^{8} + 5\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 7 + 10\cdot 13 + 6\cdot 13^{2} + 11\cdot 13^{3} + 9\cdot 13^{4} + 2\cdot 13^{5} + 11\cdot 13^{6} + 9\cdot 13^{7} + 7\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 9 + 9\cdot 13 + 2\cdot 13^{2} + 11\cdot 13^{3} + 5\cdot 13^{4} + 6\cdot 13^{5} + 4\cdot 13^{6} + 2\cdot 13^{7} + 3\cdot 13^{8} + 8\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 10 + 12\cdot 13 + 9\cdot 13^{2} + 4\cdot 13^{3} + 4\cdot 13^{4} + 11\cdot 13^{5} + 6\cdot 13^{6} + 13^{7} + 13^{8} + 6\cdot 13^{9} +O(13^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 11 + 9\cdot 13 + 6\cdot 13^{2} + 8\cdot 13^{3} + 13^{4} + 2\cdot 13^{5} + 11\cdot 13^{6} + 9\cdot 13^{7} + 7\cdot 13^{8} + 5\cdot 13^{9} +O(13^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,7)(3,4)(5,6)$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.