Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(5187\)\(\medspace = 3 \cdot 7 \cdot 13 \cdot 19 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.2.8968323.2 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.5187.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{1729})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} + 23x^{2} - 300 \)
|
The roots of $f$ are computed in $\Q_{ 37 }$ to precision 6.
Roots:
| $r_{ 1 }$ | $=$ |
\( 14 + 16\cdot 37 + 29\cdot 37^{2} + 7\cdot 37^{3} + 9\cdot 37^{4} + 27\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 2 }$ | $=$ |
\( 15 + 34\cdot 37 + 37^{2} + 10\cdot 37^{3} + 6\cdot 37^{4} + 2\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 3 }$ | $=$ |
\( 22 + 2\cdot 37 + 35\cdot 37^{2} + 26\cdot 37^{3} + 30\cdot 37^{4} + 34\cdot 37^{5} +O(37^{6})\)
|
| $r_{ 4 }$ | $=$ |
\( 23 + 20\cdot 37 + 7\cdot 37^{2} + 29\cdot 37^{3} + 27\cdot 37^{4} + 9\cdot 37^{5} +O(37^{6})\)
|
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.