Entry 64

Define \(d_k=\eta(k\tau)\) with the Dedekind eta function \(\eta(\tau)\) then for Level 42 $$\begin{align}
a &= \left(\frac{d_1^2\,d_{14}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right), \quad b = \left(\frac{d_2^2\,d_{7}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right)\\
c &= \left( \frac{d_3^2\,d_{42}^2}{d_1\,d_6\,d_{14}\,d_{21}}\right), \quad d = \left(\frac{d_6^2\,d_{21}^2}{d_2\,d_3\,d_{7}\,d_{42}}\right)\end{align}$$ The quadruple \((a,b,c,d)\) obey $$\begin{align}a\, b\, c\, d &=1\\ a+3 &= b\\ c+1 &= d\end{align}$$ and their ratios are squares$$\begin{align}\frac{a}{d} &= \left(\frac{d_1\,d_{14}}{d_6\,d_{21}}\right)^2\\ \frac{b}{c} &= \left(\frac{d_2\,d_{7}}{d_3\,d_{42}}\right)^2\end{align}$$