Consider an elliptic curve $X=\mathbf{C}/ (\mathbf{Z}+\tau \mathbf{Z})$, where $\tau$ is an element in the complex upper half plane. We define $$\Vert \Delta\Vert(X) = (\Im \tau)^6 \vert q\prod_{k=1}^\infty (1-q^k)^{24}\vert,$$ where we write $q=\exp(2\pi i \tau)$ as usual. This is called the modular discriminant of $X$.

Assume $j$ is algebraic, i.e., $X$ can be defined over a number field.

**Question.** Can the function $\log\Vert \Delta \Vert(-)$ (on the moduli space of elliptic curves over $\mathbf{C}$) be bounded (from above or below) in terms of the (height of the) $j$-invariant?

Firstly, one should be able to answer this question **ineffectively**. That is, to give a **yes** or **no** answer to the question. An effective bound (if it exists) might be a bit harder to obtain.

I heavily edited this old question. Therefore, the first four comments below might not make sense anymore.