Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(579\)\(\medspace = 3 \cdot 193 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.1737.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.579.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{193})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} - 5x^{2} + 6x + 12 \)
|
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 47 + 19\cdot 67 + 14\cdot 67^{2} + 30\cdot 67^{3} + 64\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 48 + 32\cdot 67 + 26\cdot 67^{2} + 28\cdot 67^{3} + 58\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 50 + 57\cdot 67 + 7\cdot 67^{2} + 36\cdot 67^{3} + 44\cdot 67^{4} +O(67^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 57 + 23\cdot 67 + 18\cdot 67^{2} + 39\cdot 67^{3} + 33\cdot 67^{4} +O(67^{5})\)
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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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,3)$ | $0$ |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.