Eta-products and modular elliptic curves 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 $(E,F)$ ?
 A: This won't fit into a comment box so here it is :
I came across another small nugget in Kilford's book  (p.101) this afternoon.  Let $k>0$, $N>0$ be integers such that $k.(N+1)=24$, namely one of the pairs 
$(12,1)$, $(8,2)$, $(6,3)$, $(4,5)$, $(3,7)$, $(2,11)$, or $(1,23)$.  
Then $(\eta(q)\eta(q^N))^k$ is in $S_k(\Gamma_0(N))$ unless $k=1$ or $k=3$, in which case it is in $S_k(\Gamma_0(N),({{}\over N}))$.  This implies in particular that the form 
$q\prod_{n>0}(1-q^n)(1-q^{23n})$ 
which appears in Emerton's answer  belongs to $S_1(\Gamma_0(23),({{}\over 23}))$.  
A: There is an exhaustive list in the paper [Y. Martin and K. Ono, Eta-Quotients and Elliptic
Curves, Proc. Amer. Math Soc. 125 (1997), no. 11, 3169--3176]. 
Suppose that $E_N$ is an elliptic curve of conductor
$N$, then the corresponding $L$-series is assigned to the eta product
$$
\eta(a\tau)\eta(ab\tau)\eta(ac\tau)\eta(abc\tau),
$$
where $a+ab+ac+abc=24$, $a,b,c\in\mathbb Z$,
for the following values of $N$ and $(b,c)$:
$$    \begin{align*}
        N &\quad (b,c)\cr
        11 &\quad (1,11)\cr
        14 &\quad (2,7)\cr
        15 &\quad (3,5)\cr
        20 &\quad (1,5)\cr
        24 &\quad (2,3)\cr
        27 &\quad (1,3)\cr
        32 &\quad (1,2)\cr
        36 &\quad (1,1)\cr
    \end{align*}
$$
It is probably more exciting that all these elliptic curves and their $L$-series at $s=2$ appear in Boyd's conjectures on Mahler's measure. For a nice review of this story see
[M.D. Rogers, Hypergeometric formulas for lattice sums and Mahler measures, arXiv:0806.3590] and the original paper [D.W. Boyd, Mahler's measure and special values of $L$-functions, Experiment. Math. 7 (1998) 37--82].
A: 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.
