Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(1152\)\(\medspace = 2^{7} \cdot 3^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.4.27648.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.8.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 6x^{2} + 3 \)
|
The roots of $f$ are computed in $\Q_{ 73 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 23 + 20\cdot 73 + 65\cdot 73^{2} + 69\cdot 73^{3} + 39\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 34 + 28\cdot 73 + 25\cdot 73^{2} + 64\cdot 73^{3} + 16\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 39 + 44\cdot 73 + 47\cdot 73^{2} + 8\cdot 73^{3} + 56\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 52\cdot 73 + 7\cdot 73^{2} + 3\cdot 73^{3} + 33\cdot 73^{4} +O(73^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 4 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 4 }$ | Character value | Complex conjugation |
| $1$ | $1$ | $()$ | $2$ | ✓ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |