This thesis consists of three parts. The first part provides closed formulas for $\ell$ nontrivial coefficients of the classical modular polynomial $\Phi_{\ell}(X,Y)$ in terms of the Fourier coefficients of the modular invariant function $j(z)$ for a prime $\ell$. From these formulas, we deduce results that support our conjectures on congruences modulo powers of the primes $2,3$ and $5$ satisfied by the coefficients of the classical modular polynomials. The second part focuses on the Kodaira types of elliptic curves with potentially good supersingular reduction. Our study of the Kodaira types under isogenies of elliptic curves partly motivates the first part of the thesis. The third part investigates rational spherical triangles which are triangles on the unit sphere such that the lengths of their sides and their area are rational multiples of $\pi$. We introduce a conjecture regarding the solutions to a trigonometric Diophantine equation. An implication of the conjecture determines all the rational spherical triangles. We prove some partial results towards the conjecture.