Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(384\)\(\medspace = 2^{7} \cdot 3 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.3072.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.24.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{3})\) |
Defining polynomial
$f(x)$ | $=$ |
\( x^{4} - 2x^{2} + 3 \)
|
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 2 + 5\cdot 11 + 8\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)
$r_{ 2 }$ |
$=$ |
\( 3 + 2\cdot 11 + 2\cdot 11^{2} + 4\cdot 11^{4} +O(11^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 8 + 8\cdot 11 + 8\cdot 11^{2} + 10\cdot 11^{3} + 6\cdot 11^{4} +O(11^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 9 + 5\cdot 11 + 10\cdot 11^{2} + 2\cdot 11^{3} + 5\cdot 11^{4} +O(11^{5})\)
| |
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.