Modular Elliptic Curves and Eta Products

Yes, there are a few elliptic curves whos modular forms exhibit such nice factorizations in terms of the $eta$-function. One example that I (re)found a while ago is

$$E/\mathbb{Q}:\,y^2=x^3+1$$

which has the associated modular form

$$\eta^4(6\tau).$$

You can view more of these here. And while trying to find that PDF again, I also found this had been asked previously on math overflow.

For several values of $\lambda\in \Bbb Q\setminus \{0,1\}$ the elliptic curves

$$E_{\lambda}\colon y^2=x(x-1)(x-\lambda)$$ correspond to modular forms which are linear combinations of eta-quotients, see here. For more details see the MO question, also linked by $dx dy dz$.