Basic invariants
| Dimension: | $2$ |
| Group: | $Q_8:C_2$ |
| Conductor: | \(2304\)\(\medspace = 2^{8} \cdot 3^{2} \) |
| Artin number field: | Galois closure of 8.0.3057647616.2 |
| Galois orbit size: | $2$ |
| Smallest permutation container: | $Q_8:C_2$ |
| Parity: | odd |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{-3})\) |
Galois action
Roots of defining polynomial
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 11 + 22\cdot 73 + 28\cdot 73^{2} + 3\cdot 73^{3} + 47\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 17 + 25\cdot 73 + 69\cdot 73^{2} + 43\cdot 73^{3} + 23\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 58\cdot 73 + 33\cdot 73^{2} + 72\cdot 73^{3} + 51\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 33 + 48\cdot 73 + 17\cdot 73^{2} + 27\cdot 73^{3} + 29\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 40 + 24\cdot 73 + 55\cdot 73^{2} + 45\cdot 73^{3} + 43\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 51 + 14\cdot 73 + 39\cdot 73^{2} + 21\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 7 }$ | $=$ |
\( 56 + 47\cdot 73 + 3\cdot 73^{2} + 29\cdot 73^{3} + 49\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 8 }$ | $=$ |
\( 62 + 50\cdot 73 + 44\cdot 73^{2} + 69\cdot 73^{3} + 25\cdot 73^{4} +O(73^{5})\)
|
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 values | |
| $c1$ | $c2$ | |||
| $1$ | $1$ | $()$ | $2$ | $2$ |
| $1$ | $2$ | $(1,8)(2,7)(3,6)(4,5)$ | $-2$ | $-2$ |
| $2$ | $2$ | $(1,4)(2,3)(5,8)(6,7)$ | $0$ | $0$ |
| $2$ | $2$ | $(3,6)(4,5)$ | $0$ | $0$ |
| $2$ | $2$ | $(1,6)(2,4)(3,8)(5,7)$ | $0$ | $0$ |
| $1$ | $4$ | $(1,7,8,2)(3,4,6,5)$ | $-2 \zeta_{4}$ | $2 \zeta_{4}$ |
| $1$ | $4$ | $(1,2,8,7)(3,5,6,4)$ | $2 \zeta_{4}$ | $-2 \zeta_{4}$ |
| $2$ | $4$ | $(1,6,8,3)(2,4,7,5)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,5,8,4)(2,6,7,3)$ | $0$ | $0$ |
| $2$ | $4$ | $(1,7,8,2)(3,5,6,4)$ | $0$ | $0$ |