  Recently, Hong, Mertens, Ono and Zhang proved a conjecture of
C\u{a}ld\u{a}raru, He, and Huang that expresses the Taylor series of the
modular $j$-function around the elliptic points $i$ and $\rho=e^{\pi i/3}$ as
rational functions arising from the signature 2 and 3 cases of Ramanujan's
theory of elliptic functions to alternative bases. We extend these results and
give inversion formulas for the $j$-function around $i$ and $\rho$ arising from
Gauss' hypergeometric functions and Ramanujan's theory in signatures 4 and 6.
