The modular equation of degree n gives an algebraic connection of the form

between the transcendental complete elliptic integrals of the first kind with moduli k and l. When k and l satisfy a modular equation, a relationship of the form

exists, and M is called the multiplier. In general, if p is an odd prime, then the modular equation is given by

where

is a elliptic lambda function, and

(Borwein and Borwein 1987, p. 126). An elliptic integral identity gives

so the modular equation of degree 2 is

which can be written as

A few low order modular equations written in terms of k and l are

	(9)
	(10)
	(11)

In terms of u and v,

where

and

Here, are Jacobi theta functions.

A modular equation of degree for can be obtained by iterating the equation for . Modular equations for prime p from 3 to 23 are given in Borwein and Borwein (1987).

Quadratic modular identities include

Cubic identities include

A seventh-order identity is

From Ramanujan (1913-1914),

When k and l satisfy a modular equation, a relationship of the form

exists, and M is called the multiplier. The multiplier of degree n can be given by

where is a Jacobi theta function and K(k) is a complete elliptic integral of the first kind.

The first few multipliers in terms of l and k are

In terms of the u and v defined for modular equations,

 

Modular Form, Modular Function, Schläfli's Modular Form