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 $(E,F)$ ?
Best Answer
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].