TITLE: Eta-products and modular elliptic curves
QUESTION [17 upvotes]: 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 $(E,F)$ ?

REPLY [2 votes]: Today I came across the following theorem of Mersmann: There are precisely 14 primitive eta-products which are holomorphic modular forms of weight $\frac{1}{2}$, namely...
He also proves a conjecture of Zagier to the effect that there are essentially only finitely many such products of any given weight.
I learnt these facts from Zagier's contribution to the The 1-2-3 of modular forms (which happens to have been metioned by Wadim Zudilin).
Addendum (2011/02/10)  Came across an advertisement for the book Eta products and theta series identities by Günter Köhler.