Ramanujan's continued fractions of order $10$ as modular functions

We explore the modularity of the continued fractions I(τ), J(τ), T₁(τ), T₂(τ) and U(τ)=I(τ)/J(τ) of order $10$, where I(τ) and J(τ) are introduced by Rajkhowa and Saikia, which are special cases of certain identities of Ramanujan. In particular, we show that these fractions can be expressed in terms of an η-quotient g(τ) that generates the field of all modular functions on the congruence subgroup Γ₀(10). Consequently, we prove that modular equations for g(τ) and U(τ) exist at any level, and derive these equations of prime levels p≤ 11. We also show that the above continued fractions of order $10$ can be explicitly evaluated using a singular value of g(τ), which, under certain conditions, generates the Hilbert class field of an imaginary class field. We employ the methods of Lee and Park to establish our results.