# Modular Elliptic Curves and Eta Products

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?

• 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. – reuns Sep 6 '19 at 19:45

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.

• Ah thank you for the link, it is exactly what I'm looking for. – Kevin Sep 6 '19 at 14:54
• Do you mean the MO-link or the first link with the paper by Ono? – Dietrich Burde Sep 6 '19 at 14:54
• @DietrichBurde Actually both, the MO question is weirdly similar to mine, even the conductor is the same. – Kevin Sep 6 '19 at 14:55

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 $$dx dy dz$$.