Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(1024\)\(\medspace = 2^{10} \) |
| Artin number field: | Galois closure of 8.2.2147483648.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Projective image: | $D_4$ |
| Projective field: | Galois closure of 4.2.2048.1 |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 7.
Roots:
| $r_{ 1 }$ | $=$ |
\( 5 + 57\cdot 73 + 16\cdot 73^{2} + 6\cdot 73^{3} + 3\cdot 73^{4} + 10\cdot 73^{5} + 61\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 2 }$ | $=$ |
\( 11 + 47\cdot 73 + 29\cdot 73^{2} + 52\cdot 73^{3} + 27\cdot 73^{4} + 36\cdot 73^{5} + 30\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 3 }$ | $=$ |
\( 23 + 16\cdot 73 + 43\cdot 73^{2} + 25\cdot 73^{3} + 66\cdot 73^{4} + 8\cdot 73^{5} + 58\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 4 }$ | $=$ |
\( 36 + 31\cdot 73 + 24\cdot 73^{2} + 67\cdot 73^{3} + 25\cdot 73^{4} + 2\cdot 73^{5} + 60\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 5 }$ | $=$ |
\( 37 + 41\cdot 73 + 48\cdot 73^{2} + 5\cdot 73^{3} + 47\cdot 73^{4} + 70\cdot 73^{5} + 12\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 6 }$ | $=$ |
\( 50 + 56\cdot 73 + 29\cdot 73^{2} + 47\cdot 73^{3} + 6\cdot 73^{4} + 64\cdot 73^{5} + 14\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 7 }$ | $=$ |
\( 62 + 25\cdot 73 + 43\cdot 73^{2} + 20\cdot 73^{3} + 45\cdot 73^{4} + 36\cdot 73^{5} + 42\cdot 73^{6} +O(73^{7})\)
|
| $r_{ 8 }$ | $=$ |
\( 68 + 15\cdot 73 + 56\cdot 73^{2} + 66\cdot 73^{3} + 69\cdot 73^{4} + 62\cdot 73^{5} + 11\cdot 73^{6} +O(73^{7})\)
|
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 | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $4$ | $2$ | $(1,7)(2,8)(4,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,2,8,7)(3,4,6,5)$ | $0$ | $0$ |
| $4$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $8$ | $(1,6,2,5,8,3,7,4)$ | $-\zeta_{8}^{3} - \zeta_{8}$ | $\zeta_{8}^{3} + \zeta_{8}$ |
| $2$ | $8$ | $(1,3,2,4,8,6,7,5)$ | $\zeta_{8}^{3} + \zeta_{8}$ | $-\zeta_{8}^{3} - \zeta_{8}$ |