Modular equations for some $\eta $-products
The classical modular equations involve bivariate polynomials that can be seen to be univariate in the modular invariant $j$ with integer coefficients. Kiepert found modular equations relating some $\eta $-quotients and the Weber functions $\gamma _2$ and $\gamma _3$. In the present work, we extend this idea to double $\eta $-quotients and characterize all the parameters leading to this kind of equation. We give some properties of these equations, explain how to compute them and give numerical examples.