Recently the elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ (which appears in [my answer][1]) 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 asssociated to the isogenous elliptic curve $y^2+y=x^3-x^2-10x-20$ which appears in [Franz's question][2].)

**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][3]

and 

--- [p.18][4] of Ono's *Web of modularity* on $\eta$-quotients.

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

  [1]: http://mathoverflow.net/questions/11747/galoisian-sets-of-prime-numbers/32939#32939
  [2]: http://mathoverflow.net/questions/32964/quaternary-quadratic-forms-and-elliptic-curves-via-langlands
  [3]: http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.nmj/1118787564
  [4]: http://books.google.com/books?id=MLdRYIg6pDkC&lpg=PR1&dq=ono%2520web%2520of%2520modularity&pg=PA18#v=onepage&q&f=false