We shall explore the geometry of the Modular curve $X_0(p^2)$ and it's regular minimal model over the ring of integers, which is an arithmetic surface. After a base change we shall show that the regular minimal model is semi-simple. Arakelov has introduced an intersection pairing for divisors on arithmetic surfaces. We shall derive an expression for the Arakelov self-intersection of the relative dualising sheaf on the regular minimal model of $X_0(p^2)$. As a consequence, we shall give some number theoretic applications for this computation. This is a joint work with Debargha Banerjee and Diganta Borah