Cohomology of union of quadric surfaces in $\mathbb{C}P^3$

Question

It is known that a degree 4 elliptic curve $E\subset \mathbb{C}P^3$ is the complete intersection of two irreducible quadric surfaces $E=Q_1 \cap Q_2.$ Can one compute the (co)homology groups (over $\mathbb{Q}$ coefficients) of the union of these quadric surfaces $H^*(Q_1 \cup Q_2)?$
E.g. by looking at the Mayer–Vietoris sequence, one can get that $H^0=\mathbb{Q},H^1=0,H^4=\mathbb{Q}^2,$ but two results $$H^2=\mathbb{Q}^6, H^3=\mathbb{Q}$$ and $$H^2=\mathbb{Q}^5, H^3=0$$ both fit in it (at least algebraically). Is there a way to find out which one is correct?