Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(768\)\(\medspace = 2^{8} \cdot 3 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.6144.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| 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
| $f(x)$ | $=$ |
\( x^{4} + 4x^{2} + 6 \)
|
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 15\cdot 19 + 16\cdot 19^{2} + 15\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 7 + 5\cdot 19 + 11\cdot 19^{2} + 2\cdot 19^{3} +O(19^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 12 + 13\cdot 19 + 7\cdot 19^{2} + 16\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 + 3\cdot 19 + 2\cdot 19^{2} + 3\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |