Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2016\)\(\medspace = 2^{5} \cdot 3^{2} \cdot 7 \) |
| Artin stem field: | Galois closure of 8.0.1792336896.7 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.56.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-6}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 9x^{6} + 45x^{4} - 72x^{2} + 36 \)
|
The roots of $f$ are computed in $\Q_{ 79 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 16 + 77\cdot 79 + 54\cdot 79^{2} + 26\cdot 79^{3} + 48\cdot 79^{4} + 30\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 24 + 58\cdot 79 + 55\cdot 79^{2} + 16\cdot 79^{3} + 68\cdot 79^{4} + 69\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 37 + 17\cdot 79 + 3\cdot 79^{2} + 49\cdot 79^{3} + 44\cdot 79^{4} + 33\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 39 + 75\cdot 79 + 46\cdot 79^{2} + 46\cdot 79^{3} + 49\cdot 79^{4} + 5\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 + 3\cdot 79 + 32\cdot 79^{2} + 32\cdot 79^{3} + 29\cdot 79^{4} + 73\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 6 }$ | $=$ |
\( 42 + 61\cdot 79 + 75\cdot 79^{2} + 29\cdot 79^{3} + 34\cdot 79^{4} + 45\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 7 }$ | $=$ |
\( 55 + 20\cdot 79 + 23\cdot 79^{2} + 62\cdot 79^{3} + 10\cdot 79^{4} + 9\cdot 79^{5} +O(79^{6})\)
|
| $r_{ 8 }$ | $=$ |
\( 63 + 79 + 24\cdot 79^{2} + 52\cdot 79^{3} + 30\cdot 79^{4} + 48\cdot 79^{5} +O(79^{6})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ |
| $2$ | $2$ | $(1,3)(2,5)(4,7)(6,8)$ | $0$ |
| $2$ | $2$ | $(1,8)(2,7)$ | $0$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $-2 \zeta_{4}$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $2 \zeta_{4}$ |
| $2$ | $4$ | $(1,3,8,6)(2,5,7,4)$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.