Level 20

I. Monster functions: Let,
$$a(\tau) =  \left(\frac{d_1\, d_4\, d_{10}}{d_2\,  d_5\, d_{20}}\right)^{2},\quad b(\tau) =  \frac{d_1\,d_4\,d_{10}^{10}}{d_2^2\,d_5^5\,d_{20}^5}, \quad c(\tau) =  \frac{d_2^{8}}{d_1^3\,d_4^3\,d_5\,d_{20}}$$
such that \(a+1=b,\; a+5=c\). This level 20 triple can be derived from a level 10 triple. Then define,
$$\begin{align}
j_{20A}(\tau) &= \left(\frac{d_2^2\, d_{10}^2}{d_1\,  d_4\, d_5\, d_{20}}\right)^4=\frac{bc}{a} \\
&= \left( \sqrt{j_{20F}}+\frac1{\sqrt{ j_{20F}}}\right)^2\\
j_{20B}(\tau) &=\sqrt{j_{10A}(2\tau)}\\
&= j_{20D}(\tau)+\frac4{j_{20D}(\tau)}\\
&= j_{20E}(\tau)-\frac1{j_{20E}(\tau)} \\
\color{red}{j_{20C}}(\tau) &= \left(\frac{d_1\, d_4\, d_{10}}{d_2\,  d_5\, d_{20}}\right)^{2}\;\;=\;\;a\\
j_{20D}(\tau) &= \left(\frac{d_2\, d_{10}}{d_4\,  d_{20}}\right)^{2}=\sqrt{j_{10B}(2\tau)}\\
j_{20E}(\tau) &= \left(\frac{d_4\, d_{10}}{d_2\,  d_{20}}\right)^{3}=\sqrt{j_{10D}(2\tau)}\\
j_{20F}(\tau) &= \left(\frac{d_4\, d_5}{d_1\,  d_{20}}\right)^{2}
\end{align}$$
Examples: \(\; j_{10A}\left(\tfrac12\sqrt{-19/10}\right) = 76^2,\;\; j_{20B}\left(\tfrac14\sqrt{-19/10}\right) = 76\)

II. Non-monster functions. Also define,
$$\begin{align}
u(\tau) &= \left(\frac{d_2^4\,d_5\,d_{20}}{d_1\,d_4\,d_{10}^4}\right)^2 = \frac{ac}b\\
v(\tau) &= \left(\frac{d_1\,d_4\,d_{10}^2}{d_2^2\,d_5\,d_{20}}\right)^6 =\frac{ab}c
\end{align}$$
III. Relations. Like \(j_{30B}\), the function \(j_{20A}\) is expressible by others in 4 ways. There is no linear relation between purely monster functions of level 20.