Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Artin stem field: | Galois closure of 8.2.36691771392.4 |
| 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 $\Q_{ 109 }$ to precision 10.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 61\cdot 109 + 42\cdot 109^{2} + 65\cdot 109^{3} + 26\cdot 109^{4} + 56\cdot 109^{5} + 69\cdot 109^{6} + 4\cdot 109^{7} + 35\cdot 109^{8} + 5\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 100\cdot 109 + 91\cdot 109^{2} + 69\cdot 109^{3} + 104\cdot 109^{4} + 76\cdot 109^{5} + 5\cdot 109^{6} + 69\cdot 109^{7} + 43\cdot 109^{8} + 15\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 3 }$ | $=$ |
\( 10 + 19\cdot 109 + 21\cdot 109^{2} + 27\cdot 109^{3} + 3\cdot 109^{4} + 56\cdot 109^{5} + 90\cdot 109^{6} + 78\cdot 109^{7} + 62\cdot 109^{8} + 51\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 4 }$ | $=$ |
\( 43 + 54\cdot 109 + 4\cdot 109^{2} + 33\cdot 109^{3} + 97\cdot 109^{4} + 74\cdot 109^{5} + 22\cdot 109^{6} + 41\cdot 109^{7} + 108\cdot 109^{8} + 100\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 5 }$ | $=$ |
\( 66 + 54\cdot 109 + 104\cdot 109^{2} + 75\cdot 109^{3} + 11\cdot 109^{4} + 34\cdot 109^{5} + 86\cdot 109^{6} + 67\cdot 109^{7} + 8\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 6 }$ | $=$ |
\( 99 + 89\cdot 109 + 87\cdot 109^{2} + 81\cdot 109^{3} + 105\cdot 109^{4} + 52\cdot 109^{5} + 18\cdot 109^{6} + 30\cdot 109^{7} + 46\cdot 109^{8} + 57\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 7 }$ | $=$ |
\( 106 + 8\cdot 109 + 17\cdot 109^{2} + 39\cdot 109^{3} + 4\cdot 109^{4} + 32\cdot 109^{5} + 103\cdot 109^{6} + 39\cdot 109^{7} + 65\cdot 109^{8} + 93\cdot 109^{9} +O(109^{10})\)
|
| $r_{ 8 }$ | $=$ |
\( 107 + 47\cdot 109 + 66\cdot 109^{2} + 43\cdot 109^{3} + 82\cdot 109^{4} + 52\cdot 109^{5} + 39\cdot 109^{6} + 104\cdot 109^{7} + 73\cdot 109^{8} + 103\cdot 109^{9} +O(109^{10})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,3)(4,5)(6,7)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $4$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $8$ | $(1,6,5,7,8,3,4,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,3,5,2,8,6,4,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.