The complete elliptic integral of the first kind provides the k modulus which is shown to connect these integrals and functions to the modular functions previously used for calculating the class invariant and the octic q continued fraction.  The k modulus is an algebraic integer between 0 and 1.  Equations are developed for odd and even class invariants and their associated octahedron.  The connection to the Elliptic Theta functions and the j-invariant is also shown.

Elliptic Functions and the Ramanujan Octave