Ramanujan's continued fractions of order 10 as modular functions

We explore the modularity of the continued fractions $I\left(\tau \right),J\left(\tau \right),T_1\left(\tau \right),T_2\left(\tau \right)$, and $U\left(\tau \right)=I\left(\tau \right)/J\left(\tau \right)$ of order 10, which are special cases of certain identities of Ramanujan. The continued fractions $I\left(\tau \right)$ and $J\left(\tau \right)$ were recently introduced by Rajkhowa and Saikia. We show that these continued fractions can be expressed in terms of an η-quotient $g\left(\tau \right)$ that generates the field of all modular functions on the congruence subgroup ${\Gamma }_0\left(10\right)$. Consequently, we deduce that the modular equations for $g\left(\tau \right)$ and $U\left(\tau \right)$ exist at any level and derive these equations of prime levels $p\le 11$. We also show that the continued fractions of order 10 can be explicitly evaluated using a singular value of $g\left(\tau \right)$, which under certain conditions generates the Hilbert class field of an imaginary quadratic field. We employ the methods of Lee and Park to establish our results.