Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(640\)\(\medspace = 2^{7} \cdot 5 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.2560.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.40.2t1.b.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{10})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - 6x^{2} + 10 \)
|
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 3 + 19\cdot 37 + 6\cdot 37^{2} + 6\cdot 37^{3} + 17\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 16 + 2\cdot 37 + 22\cdot 37^{2} + 32\cdot 37^{3} + 33\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 21 + 34\cdot 37 + 14\cdot 37^{2} + 4\cdot 37^{3} + 3\cdot 37^{4} +O(37^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 34 + 17\cdot 37 + 30\cdot 37^{2} + 30\cdot 37^{3} + 19\cdot 37^{4} +O(37^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $2$ | |
| $1$ | $2$ | $(1,4)(2,3)$ | $-2$ | |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ | ✓ |
| $2$ | $2$ | $(1,4)$ | $0$ | |
| $2$ | $4$ | $(1,3,4,2)$ | $0$ |