In this article at page $4$ F. Beukers introduces a function $$y(\tau) = \frac{\eta^{8}(6\tau)\eta^{4}(\tau)}{\eta^{8}(2\tau)\eta^{4}(3\tau)}\tag{1}$$ where $\tau$ is a complex number with positive imaginary part and Dedekind's eta function $\eta(\tau)$ is defined by $$\eta(\tau) = e^{\pi i\tau/12}\prod_{n = 1}^{\infty}(1 - e^{2\pi in\tau})\tag{2}$$ Beukers then mentions that the function $y(\tau)$ satisfies the following equation $$y\Big(\frac{-1}{6\tau}\Big) = \frac{y(\tau) - 1/9}{y(\tau) - 1}\tag{3}$$
But I can't check that. I have already tried to use the functional equation for eta function $$\eta(-1/\tau)=\sqrt{-i\tau}\;\eta(\tau)\tag{4}$$ but this doesn't work. Has this something to do with Beukers remark "$y(\tau)$ generates the field of modular functions"? But why does $y(\tau)$ generate this field?
Thanks a lot in advance.