Basic invariants
Dimension: | $2$ |
Group: | $D_{4}$ |
Conductor: | \(584\)\(\medspace = 2^{3} \cdot 73 \) |
Frobenius-Schur indicator: | $1$ |
Root number: | $1$ |
Artin stem field: | Galois closure of 4.2.42632.1 |
Galois orbit size: | $1$ |
Smallest permutation container: | $D_{4}$ |
Parity: | odd |
Determinant: | 1.584.2t1.b.a |
Projective image: | $C_2^2$ |
Projective field: | Galois closure of \(\Q(\sqrt{-2}, \sqrt{73})\) |
Defining polynomial
$f(x)$ | $=$ | \( x^{4} + x^{2} - 18 \) . |
The roots of $f$ are computed in $\Q_{ 19 }$ to precision 5.
Roots:
$r_{ 1 }$ | $=$ | \( 7 + 14\cdot 19 + 5\cdot 19^{2} + 13\cdot 19^{3} +O(19^{5})\) |
$r_{ 2 }$ | $=$ | \( 8 + 4\cdot 19 + 10\cdot 19^{2} + 9\cdot 19^{3} + 7\cdot 19^{4} +O(19^{5})\) |
$r_{ 3 }$ | $=$ | \( 11 + 14\cdot 19 + 8\cdot 19^{2} + 9\cdot 19^{3} + 11\cdot 19^{4} +O(19^{5})\) |
$r_{ 4 }$ | $=$ | \( 12 + 4\cdot 19 + 13\cdot 19^{2} + 5\cdot 19^{3} + 18\cdot 19^{4} +O(19^{5})\) |
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.