  We study the stratum in the set of all quadratic differential systems
$\dot{x}=P_2(x,y), \dot{y}=Q_2(x,y)$ with a center, known as the
codimension-four case $Q_4$. It has a center and a node and a rational first
integral. The limit cycles under small quadratic perturbations in the system
are determined by the zeros of the first Poincar\'e-Pontryagin-Melnikov
integral $I$. We show that the orbits of the unperturbed system are elliptic
curves, and $I$ is a complete elliptic integral. Then using Picard-Fuchs
equations and the Petrov's method (based on the argument principle), we set an
upper bound of eight for the number of limit cycles produced from the period
annulus around the center.
