Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(900\)\(\medspace = 2^{2} \cdot 3^{2} \cdot 5^{2} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.13500.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.4.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(i, \sqrt{15})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 15 \)
|
The roots of $f$ are computed in $\Q_{ 113 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 35 + 25\cdot 113 + 78\cdot 113^{2} + 31\cdot 113^{3} + 91\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 40 + 112\cdot 113 + 88\cdot 113^{2} + 28\cdot 113^{3} + 28\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 73 + 24\cdot 113^{2} + 84\cdot 113^{3} + 84\cdot 113^{4} +O(113^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 78 + 87\cdot 113 + 34\cdot 113^{2} + 81\cdot 113^{3} + 21\cdot 113^{4} +O(113^{5})\)
|
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.