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By introducing a class of meromorphic functions with certain ramificationstructures on $\Bbb{CP}^1$, a new method for the determination of the Legendrerepresentation of elliptic curves with complex multiplication is introduced.These functions reduce the desired representation to the solution of anexplicitly given system of polynomial equations and makes no use of theknowledge of Hilbert class polynomial on which standard computations depend. Asa byproduct, an algorithm for computing $j(k\tau)$ in terms of $j(\tau)$ isobtained and implemented for $k=2,3$. Solving the system of equations providesa method for computing certain special values of the modular function$j:\Bbb{H}\rightarrow\Bbb{C}$.

On the $j$-invariant and the Legendre Representation of Elliptic Curves with Complex Multiplication