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 4.0.3072.2 |
| 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 $\Q_{ 59 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 9 + 2\cdot 59 + 35\cdot 59^{2} + 26\cdot 59^{3} + 23\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 25 + 25\cdot 59 + 44\cdot 59^{2} + 17\cdot 59^{3} + 12\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 34 + 33\cdot 59 + 14\cdot 59^{2} + 41\cdot 59^{3} + 46\cdot 59^{4} +O(59^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 50 + 56\cdot 59 + 23\cdot 59^{2} + 32\cdot 59^{3} + 35\cdot 59^{4} +O(59^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,4)$ | $0$ |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.