Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(4645\)\(\medspace = 5 \cdot 929 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.23225.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | even |
| Determinant: | 1.4645.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{5}, \sqrt{929})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 15x^{2} - 8x + 64 \)
|
The roots of $f$ are computed in $\Q_{ 29 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 2 + 8\cdot 29 + 29^{2} + 24\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 4 + 13\cdot 29 + 3\cdot 29^{2} + 4\cdot 29^{3} + 29^{4} +O(29^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 5 + 18\cdot 29 + 9\cdot 29^{2} + 27\cdot 29^{3} + 13\cdot 29^{4} +O(29^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 19 + 18\cdot 29 + 14\cdot 29^{2} + 2\cdot 29^{3} +O(29^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $2$ | $2$ | $(1,3)(2,4)$ | $0$ |
| $2$ | $2$ | $(1,2)$ | $0$ |
| $2$ | $4$ | $(1,4,2,3)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.