infinite products of rational functions of Klein’s j-invariant

As a Riemann surface, the fundamental region has genus 0, and every (level one) modular function is a rational function in j; and, conversely, every rational function in j is a modular function

Well then, doesn’t that mean that:

\Lambda(\tau) = \prod_{n=1}^{\infty} \frac{j(\tau)^{2n}-3j(\tau) +1}{j(\tau)^{3n}-i j(\tau)^{n} +1} is also a modular function, each of its factors being a rational function of j(\tau). I’ll attempt to make a picture of \Lambda(\tau) later, my computers are busy doing other things


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