Is $\eta^{24}(\tau)\,j(\tau) = {E_4}^3(q)$?

Question

Given the j-function $j(\tau)$,

$j(\tau) = 1728J(\tau)$,

where $J(\tau)$ is Klein’s absolute invariant, the Dedekind eta function $\eta(\tau)$, and the following Eisenstein series,

$\begin{align} E_4 (q) &= 1+240\sum_{n=1}^{\infty} \frac{n^3q^n}{1-q^n}\\[1.5mm] E_6 (q) &= 1-504\sum_{n=1}^{\infty} \frac{n^5q^n}{1-q^n}\\[1.5mm] E_8 (q) &= 1+480\sum_{n=1}^{\infty} \frac{n^7q^n}{1-q^n}\\ \end{align}$

where,

$q = \exp(2\pi i \tau)$

Are the following relations true?:

$\begin{align} 1.\;\; \eta^{24}(\tau) &= \frac{{E_4}^3(q)}{j(\tau)}\\[1.5mm] 2.\;\; \eta^{24}(\tau) &= \frac{{E_6}^2(q)}{j(\tau)-1728}\\[1.5mm] 3.\;\; \eta^{48}(\tau) &= \frac{{E_8}^3(q)}{j^2(\tau)}\\ \end{align}$

Answer 1

Yes, these three relations are all true. Since $\eta^{24}$ is the weight 12 level 1 cusp form $\Delta$, you can write them as relations between level 1 modular forms, and these are easy to check because the relevant modular form spaces have small finite dimensions.