Following the P-Q modular equations used by B. Berndt (Transactions of the American Mathematical Society, 349(6), June 1997) to determine 13 radical forms of class invariants reported by Ramanujan, this Chapter discusses a new method of analysis. By using the q octic continued fraction and the q cubic solution equation (qkQ) derived in Chapter 28, all 13 radical forms are solved using results from Berndt’s two theorems. The radical forms for discriminants with prime divisors of 5 and 7 are presented without the need for modular P-Q equations.

A theorem of existence of a solution of the class invariant in radicals is presented based on an invariant of the class invariant and modulus of the octic q continued fraction.

expressing the octic q continued fraction in radical form using a q cubic solution equation