Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(2279\)\(\medspace = 43 \cdot 53 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.97997.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.2279.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-43}, \sqrt{53})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} + 9x^{2} + x + 1 \) . |
The roots of $f$ are computed in $\Q_{ 11 }$ to precision 6.
Roots:
$r_{ 1 }$ | $=$ |
\( 1 + 6\cdot 11 + 7\cdot 11^{2} + 2\cdot 11^{3} + 6\cdot 11^{4} + 2\cdot 11^{5} +O(11^{6})\)
$r_{ 2 }$ |
$=$ |
\( 4 + 4\cdot 11 + 4\cdot 11^{2} + 5\cdot 11^{3} + 6\cdot 11^{4} + 5\cdot 11^{5} +O(11^{6})\)
| $r_{ 3 }$ |
$=$ |
\( 8 + 5\cdot 11 + 5\cdot 11^{2} + 3\cdot 11^{3} + 10\cdot 11^{4} + 11^{5} +O(11^{6})\)
| $r_{ 4 }$ |
$=$ |
\( 10 + 5\cdot 11 + 4\cdot 11^{2} + 10\cdot 11^{3} + 9\cdot 11^{4} +O(11^{6})\)
| |
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.