Modular equation for a quotient of eta functions 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.
 A: There is a simple proof using two eta product identities. If
$$ y_1(\tau) := \eta(2\tau)\eta^{9}(6\tau)\eta^{4}(\tau), \quad
 y_2(\tau) := \eta^{9}(2\tau)\eta^{4}(3\tau)\eta(6\tau), $$
then $\, y(\tau) = y_1(\tau)/y_2(\tau), \,$ and
$$ y_2(\tau) - 9y_1(\tau) = \eta(\tau)^9\eta(3\tau)\eta(6\tau)^4, \tag{1} $$
$$ y_2(\tau) -  y_1(\tau) = \eta(\tau)\eta(2\tau)^4\eta(3\tau)^9, \tag{2} $$
$$ y_3(\tau) := \frac{y_2(\tau) - 9y_1(\tau)}{y_2(\tau) -  y_1(\tau)}
     = 9\frac{y(\tau) - 1/9}{y(\tau) - 1}, \tag{3} $$
$$ y_3(\tau) =  9 \frac{\eta(\tau)^8 \eta(6\tau)^4}
{\eta(3\tau)^8 \eta(2\tau)^4}
     = y\Big(\frac{-1}{6\tau}\Big). \tag{4}$$
Equation $(1)$ is eta product identity $t_{6,13,30}$ multiplied by $\,\eta(6\tau)\,$
and equation $(2)$ is eta product identity $t_{6,13,34}$ multiplied by $\,\eta(2\tau).\,$
Both identities have many different forms and references to proofs were listed in
their entry in my Dedekind eta product identities website. The Wikipedia article Dedekind eta function links to the Wayback machine
where my website is archived.
A: Identity (8) corresponds to an identity of Ramanujan proved by Berndt. "Ramanujan Notebooks", Part IV, p. 204, Entry 51.
