Let $Q\subset\mathbb{P}^3\subset\mathbb{P}^4$ be a smooth quadric surface, and let $X = Bl_Q\mathbb{P}^4$ the blow-up of $\mathbb{P}^4$ along $Q$. Let $H$ be the pull-back of the hyperplane section of $\mathbb{P}^4$ and $E$ the exceptional divisor. I would like to compute te intersection numbers of these divisors.
For instance $H^4 = 1$. Then we should have $H^3\cdot E = 0$ because $H^3$ is the pull-back of the class of a line and $Q$ is in codimension two. Then $H^2\cdot E^2 = -2$ because $H^2$ is the pull-back of the class of a plane and $deg(Q) = 2$. Finally the first chern class of $N_{Q/\mathbb{P}^4}$ is $3$. Therefore we should have $H\cdot E^3 = H\cdot E^2\cdot E = -2\cdot 3$, and $E^3 = -2\cdot 3^2$. Summing up we should have
$$H^4 = 1, \quad H^3\cdot E = 0, \quad H^{4-i}E^i = -2\cdot 3^{i-2},\: for \: i\geq 2.$$
Is this computation correct or am I missing something?