Analogue of Ramanujan's function k (τ) for the cubic continued fraction

We study the modularity of the function u(τ) = C(τ)C(2τ), where C(τ) is Ramanujan's cubic continued fraction. It is an analogue of Ramanujan's function k(τ) = r(τ)r(2τ)2, where r(τ) is the Rogers-Ramanujan continued fraction. We first prove the modularity of u(τ) and express C(τ) and C(2τ) in terms of u(τ). Subsequently, we find modular equations of u(τ) of level n for every positive integer n by using affine models of modular curves. Finally, we demonstrate that the value of u(τ) generates the ray class field over an imaginary quadratic field modulo 2 for some τ in an imaginary quadratic field.