Basic invariants
| Dimension: | $2$ |
| Group: | $D_{4}$ |
| Conductor: | \(63\)\(\medspace = 3^{2} \cdot 7 \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 4.0.189.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $D_{4}$ |
| Parity: | odd |
| Determinant: | 1.7.2t1.a.a |
| Projective image: | $C_2^2$ |
| Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{-7})\) |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{4} - x^{3} + 2x + 1 \)
|
The roots of $f$ are computed in $\Q_{ 67 }$ to precision 5.
Roots:
| $r_{ 1 }$ | $=$ |
\( 4 + 22\cdot 67 + 31\cdot 67^{2} + 41\cdot 67^{3} + 49\cdot 67^{4} +O(67^{5})\)
| $r_{ 2 }$ |
$=$ |
\( 11 + 54\cdot 67 + 30\cdot 67^{2} + 61\cdot 67^{3} + 5\cdot 67^{4} +O(67^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 26 + 55\cdot 67 + 57\cdot 67^{2} + 24\cdot 67^{3} + 59\cdot 67^{4} +O(67^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 27 + 2\cdot 67 + 14\cdot 67^{2} + 6\cdot 67^{3} + 19\cdot 67^{4} +O(67^{5})\)
| |
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 |
| $1$ | $1$ | $()$ | $2$ |
| $1$ | $2$ | $(1,3)(2,4)$ | $-2$ |
| $2$ | $2$ | $(1,2)(3,4)$ | $0$ |
| $2$ | $2$ | $(1,3)$ | $0$ |
| $2$ | $4$ | $(1,4,3,2)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.