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We study the modularity of Ramanujan’s functionk(τ)=r(τ)r2(2τ)k(\tau )=r(\tau ){r}^{2}(2\tau ), wherer(τ)r(\tau )is the  Rogers-Ramanujan continued fraction . We first find the modular equation ofk(τ)k(\tau )of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for someτ\tauin an imaginary quadratic field, the valuek(τ)k(\tau )generates the ray class field over an imaginary quadratic field modulo 10; this is because the function  k  is a generator of the field of the modular function onΓ1(10){{\mathrm{\Gamma}}}_{1}(10). Furthermore, we suggest a rather optimal way of evaluating the singular values ofk(τ)k(\tau )using the modular equations in the following two ways: one is that ifj(τ)j(\tau )is the elliptic modular function, then one can explicitly evaluate the valuek(τ)k(\tau ), and the other is that once the valuek(τ)k(\tau )is given, we can obtain the valuek(rτ)k(r\tau )for any positive rational number  r  immediately.

Ramanujan’s function k ( τ )= r ( τ ) r 2 (2 τ ) and its modularity