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The elliptic curve $E:y^2+y=x^3-x^2$ of conductor $11$ is interesting as the associated modular form (this is over $\bf Q$) is $$ F=q\prod_{n>0}(1-q^n)^2(1-q^{11n})^2 $$ Clearly this exhibits a very nice $\eta$-product.

Is anyone aware of 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?

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  • $\begingroup$ Every weight $2$ eta product which is a modular form is the L-function of an elliptic curve. Such a formula in term of $\eta$ is useful for the BSD conjecture. Not sure if that it doesn't vanish on the upper half-plane is helpful. $\endgroup$ – reuns Sep 6 '19 at 19:45
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Yes, there are a few elliptic curves whos modular forms exhibit such nice factorizations in terms of the $eta$-function. One example that I (re)found a while ago is

$$E/\mathbb{Q}:\,y^2=x^3+1$$

which has the associated modular form

$$\eta^4(6\tau).$$

You can view more of these here. And while trying to find that PDF again, I also found this had been asked previously on math overflow.

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  • $\begingroup$ Ah thank you for the link, it is exactly what I'm looking for. $\endgroup$ – Kevin Sep 6 '19 at 14:54
  • $\begingroup$ Do you mean the MO-link or the first link with the paper by Ono? $\endgroup$ – Dietrich Burde Sep 6 '19 at 14:54
  • $\begingroup$ @DietrichBurde Actually both, the MO question is weirdly similar to mine, even the conductor is the same. $\endgroup$ – Kevin Sep 6 '19 at 14:55
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For several values of $\lambda\in \Bbb Q\setminus \{0,1\}$ the elliptic curves

$$E_{\lambda}\colon y^2=x(x-1)(x-\lambda)$$ correspond to modular forms which are linear combinations of eta-quotients, see here. For more details see the MO question, also linked by $dx dy dz$.

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