  The modularity theorem implies that for every elliptic curve $E /\mathbb{Q}$
there exist rational maps from the modular curve $X_0(N)$ to $E$, where $N$ is
the conductor of $E$. These maps may be expressed in terms of pairs of modular
functions $X(z)$ and $Y(z)$ where $X(z)$ and $Y(z)$ satisfy the Weierstrass
equation for $E$ as well as a certain differential equation. Using these two
relations, a recursive algorithm can be used to calculate the $q$ - expansions
of these parametrizations at any cusp. %These functions are algebraic over
$\mathbb{Q}(j(z))$ and satisfy modular polynomials where each of the
coefficient functions are rational functions in $j(z)$.
  Using these functions, we determine the divisor of the parametrization and
the preimage of rational points on $E$. We give a sufficient condition for when
these preimages correspond to CM points on $X_0(N)$. We also examine a
connection between the algebras generated by these functions for related
elliptic curves, and describe sufficient conditions to determine congruences in
the $q$-expansions of these objects.
