The elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ is interesting as the associated modular form (this is over $\bf Q$) is $$ F=q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ Clearly this exhibits a very nice $\eta$-product.
Is anyone aware of other elliptic curves over $\bf Q$ which have a simple minimal equation and whose associated modular form is a nice $\eta$-product or even a nice $\eta$-quotient?
On
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 $dx dy dz$.
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.