Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.3072.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.24.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 2x^{2} + 3 \)
|
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 5\cdot 11 + 8\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 3 + 2\cdot 11 + 2\cdot 11^{2} + 4\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 8 + 8\cdot 11 + 8\cdot 11^{2} + 10\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 9 + 5\cdot 11 + 10\cdot 11^{2} + 2\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |