Why is j($i$) defined as $1728$ instead of $1?$

Question

I know that the j-invariant is very important in mathematics and is related to several "almost integers" as well as monstrous moonshine. According to Wikipedia, the j-invariant is defined as the unique function on the upper half-plane of complex numbers such that it is holomorphic away from a simple pole at the cusp, j($e^{2\pi i/3}$) is $0$, and j($i$) is $1728$. Why is there an extra factor of $1728$ in the j-invariant definition?

Answer 1

COMMENT.- The $j$-invariant is a subtle notion and can be defined up to multiplicative constant according to the desired parameterization of certain modular forms. For example if you define $$j(z)=2^{12}\cdot3^3\cdot5^3\frac{G_4^3}{\Delta}(z)=1728\cdot 8000\frac{G_4^3}{\Delta}(z)$$ you get the basic fact that a meromorphic function $f$ defined over the complex upper half-plane is invariant by action of $SL(2,\mathbb Z)$ and meromorphic at $\infty$ if and only if $f$ is a rational function of $j$. While if you define, with Serre, $$j(z)=1728\frac{g_2^3}{\Delta}(z)$$ the basic fact above mentioned becomes stronger. There is not question here of to be clair because it would suppose many definitions and properties; only we say that it is basically a notion involving lattices and double periodic functions (when elliptic curves) and fastly fall on the function $\zeta$ de Riemann. So we bound to standard definitions on a lattice $\Gamma$

$$G_k(\Gamma)=\sum_{\gamma\in\Gamma}\frac{1}{\gamma^{2k}}\space\\g_2=60G_2,\space g_3=140G_3\\\Delta=g_2^3-27g_3^2$$

(Sorry for bad English).