Why is j($i$) defined as $1728$ instead of $1?$ 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?
 A: With the definition of $j$ from Wikipedia, the $j$-function has a Fourier series
$$j(τ) = q^{-1} + 744 + 196884q + 21493760q^2+\mathcal O(q^3) $$
at $\tau\to i \infty$, where $q=e^{2\pi i\tau}$. We can easily redefine $\tilde j=j/12^3$, such that $\tilde j(i)=1$, but from the above $q$-series you can see that $\tilde j$ does not have integer Fourier coefficients.
This singles out the overall factor in $j$. Another natural definition is $j-744$, which in most applications is completely equivalent.
A: 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).
