Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(3783\)\(\medspace = 3 \cdot 13 \cdot 97 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.0.11349.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.3783.2t1.a.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-3}, \sqrt{1261})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} - x^{3} - 17x^{2} + 9x + 81 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ |
\( 9 + 4\cdot 19 + 15\cdot 19^{2} + 15\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\)
$r_{ 2 }$ |
$=$ |
\( 14 + 18\cdot 19 + 5\cdot 19^{2} + 10\cdot 19^{3} + 12\cdot 19^{4} +O(19^{5})\)
| $r_{ 3 }$ |
$=$ |
\( 17 + 3\cdot 19 + 5\cdot 19^{2} + 8\cdot 19^{3} + 5\cdot 19^{4} +O(19^{5})\)
| $r_{ 4 }$ |
$=$ |
\( 18 + 10\cdot 19 + 11\cdot 19^{2} + 3\cdot 19^{3} + 14\cdot 19^{4} +O(19^{5})\)
| |
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.