Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in my answer) became my favourite elliptic over $\bf Q$ because the associated modular form $$ F=q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ is such a nice "$\eta$-product". (This modular form is also associated to the isogenous elliptic curve $y^2+y=x^3-x^2-10x-20$ which appears in Franz's question.)

**Question.** Are there 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?

I know two references which might have a bearing on the question

--- Koike's article on McKay's conjecture

and

--- p.18 of Ono's *Web of modularity* on $\eta$-quotients.

Can someone provide a partial or exhaustive list of such nice pairs $(E,F)$ ?