Analogue of Ramanujan's function k(t) for the cubic continued fraction

We study the modularity of the function u(t) = C(t)C(2t), where C(t) is Ramanujan's cubic continued fraction. It is an analogue of Ramanujan's function k(t) = r(t)r(2t)(2), where r(t) is the Rogers-Ramanujan continued fraction. We first prove the modularity of u(t) and express C(t) and C(2t) in terms of u(t). Subsequently, we find modular equations of u(t) of level n for every positive integer n by using affine models of modular curves. Finally, we demonstrate that the value of u(t) generates the ray class field over an imaginary quadratic field modulo 2 for some t in an imaginary quadratic field.