Basic invariants
| Dimension: | $2$ |
| Group: | $QD_{16}$ |
| Conductor: | \(1792\)\(\medspace = 2^{8} \cdot 7 \) |
| Artin stem field: | Galois closure of 8.2.5754585088.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $QD_{16}$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $D_4$ |
| Projective stem field: | Galois closure of 4.0.1568.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} + 4x^{6} + 4x^{4} - 4x^{2} - 7 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 8.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 9\cdot 79 + 42\cdot 79^{2} + 18\cdot 79^{3} + 36\cdot 79^{4} + 14\cdot 79^{5} + 55\cdot 79^{6} + 47\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 2 }$ | $=$ |
\( 6 + 77\cdot 79 + 44\cdot 79^{2} + 47\cdot 79^{3} + 27\cdot 79^{4} + 55\cdot 79^{5} + 57\cdot 79^{6} + 25\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 4\cdot 79 + 10\cdot 79^{2} + 5\cdot 79^{3} + 73\cdot 79^{4} + 42\cdot 79^{5} + 7\cdot 79^{6} + 13\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 73\cdot 79 + 11\cdot 79^{2} + 6\cdot 79^{3} + 24\cdot 79^{4} + 8\cdot 79^{5} + 17\cdot 79^{6} + 62\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 5 }$ | $=$ |
\( 45 + 5\cdot 79 + 67\cdot 79^{2} + 72\cdot 79^{3} + 54\cdot 79^{4} + 70\cdot 79^{5} + 61\cdot 79^{6} + 16\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 6 }$ | $=$ |
\( 71 + 74\cdot 79 + 68\cdot 79^{2} + 73\cdot 79^{3} + 5\cdot 79^{4} + 36\cdot 79^{5} + 71\cdot 79^{6} + 65\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 7 }$ | $=$ |
\( 73 + 79 + 34\cdot 79^{2} + 31\cdot 79^{3} + 51\cdot 79^{4} + 23\cdot 79^{5} + 21\cdot 79^{6} + 53\cdot 79^{7} +O(79^{8})\)
|
| $r_{ 8 }$ | $=$ |
\( 77 + 69\cdot 79 + 36\cdot 79^{2} + 60\cdot 79^{3} + 42\cdot 79^{4} + 64\cdot 79^{5} + 23\cdot 79^{6} + 31\cdot 79^{7} +O(79^{8})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $4$ | $2$ | $(2,5)(3,6)(4,7)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $0$ |
| $4$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $0$ |
| $2$ | $8$ | $(1,4,3,7,8,5,6,2)$ | $-\zeta_{8}^{3} - \zeta_{8}$ |
| $2$ | $8$ | $(1,5,3,2,8,4,6,7)$ | $\zeta_{8}^{3} + \zeta_{8}$ |
The blue line marks the conjugacy class containing complex conjugation.