Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Artin stem field: | Galois closure of 8.0.150994944.1 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Determinant: | 1.3.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{8} - 4x^{6} + 8x^{4} - 12x^{2} + 9 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 55\cdot 73 + 69\cdot 73^{2} + 53\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 15 + 67\cdot 73 + 48\cdot 73^{2} + 44\cdot 73^{3} + 57\cdot 73^{4} +O(73^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 16 + 34\cdot 73 + 19\cdot 73^{2} + 40\cdot 73^{3} + 70\cdot 73^{4} +O(73^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 35 + 46\cdot 73 + 67\cdot 73^{2} + 69\cdot 73^{3} + 37\cdot 73^{4} +O(73^{5})\)
| $r_{ 5 }$ |
$=$ |
\( 38 + 26\cdot 73 + 5\cdot 73^{2} + 3\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)
| $r_{ 6 }$ |
$=$ |
\( 57 + 38\cdot 73 + 53\cdot 73^{2} + 32\cdot 73^{3} + 2\cdot 73^{4} +O(73^{5})\)
| $r_{ 7 }$ |
$=$ |
\( 58 + 5\cdot 73 + 24\cdot 73^{2} + 28\cdot 73^{3} + 15\cdot 73^{4} +O(73^{5})\)
| $r_{ 8 }$ |
$=$ |
\( 59 + 17\cdot 73 + 3\cdot 73^{2} + 19\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)(5,6)(7,8)$ | $0$ |
| $2$ | $2$ | $(2,7)(4,5)$ | $0$ |
| $2$ | $2$ | $(1,5)(2,3)(4,8)(6,7)$ | $0$ |
| $1$ | $4$ | $(1,6,8,3)(2,5,7,4)$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,3,8,6)(2,4,7,5)$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.