Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(576\)\(\medspace = 2^{6} \cdot 3^{2} \) |
| Artin number field: | Galois closure of 8.2.573308928.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.0.432.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 19\cdot 37 + 17\cdot 37^{2} + 5\cdot 37^{3} + 16\cdot 37^{4} + 28\cdot 37^{5} + 28\cdot 37^{6} + 18\cdot 37^{7} + 28\cdot 37^{8} + 37^{9} +O(37^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 18\cdot 37 + 20\cdot 37^{2} + 5\cdot 37^{3} + 21\cdot 37^{4} + 17\cdot 37^{5} + 23\cdot 37^{6} + 3\cdot 37^{7} + 30\cdot 37^{8} + 22\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 23\cdot 37 + 28\cdot 37^{2} + 14\cdot 37^{3} + 22\cdot 37^{4} + 13\cdot 37^{5} + 4\cdot 37^{6} + 35\cdot 37^{7} + 24\cdot 37^{8} + 16\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 14 + 15\cdot 37 + 13\cdot 37^{2} + 13\cdot 37^{3} + 26\cdot 37^{4} + 17\cdot 37^{5} + 34\cdot 37^{6} + 30\cdot 37^{7} + 32\cdot 37^{8} + 8\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 23 + 21\cdot 37 + 23\cdot 37^{2} + 23\cdot 37^{3} + 10\cdot 37^{4} + 19\cdot 37^{5} + 2\cdot 37^{6} + 6\cdot 37^{7} + 4\cdot 37^{8} + 28\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 27 + 13\cdot 37 + 8\cdot 37^{2} + 22\cdot 37^{3} + 14\cdot 37^{4} + 23\cdot 37^{5} + 32\cdot 37^{6} + 37^{7} + 12\cdot 37^{8} + 20\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 30 + 18\cdot 37 + 16\cdot 37^{2} + 31\cdot 37^{3} + 15\cdot 37^{4} + 19\cdot 37^{5} + 13\cdot 37^{6} + 33\cdot 37^{7} + 6\cdot 37^{8} + 14\cdot 37^{9} +O(37^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 32 + 17\cdot 37 + 19\cdot 37^{2} + 31\cdot 37^{3} + 20\cdot 37^{4} + 8\cdot 37^{5} + 8\cdot 37^{6} + 18\cdot 37^{7} + 8\cdot 37^{8} + 35\cdot 37^{9} +O(37^{10})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,2)(4,5)(7,8)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,4,2,6,8,5,7,3)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,5,2,3,8,4,7,6)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |