Modular Function Multiplier

When $k$ and $l$ satisfy a Modular Equation, a relationship of the form

$${M(l,k)\,dy\over \sqrt{(1-y^2)(1-l^2y^2)}} = {dx\over \sqrt{(1-x^2)(1-k^2x^2)}}$$ (1)

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

$$M_n(l,k)\equiv {{\vartheta _3}^2(q)\over {\vartheta _3}^2(q^{1/p})} = {K(k)\over K(l)},$$ (2)

where $\vartheta_i$ is a 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

$$M_2(l,k) = {1\over 1+k}={1+l'\over 2}$$ (3)

$$M_3(l,k) = {1-\sqrt{l^3\over k}\over 1-\sqrt{k^3\over l}}.$$ (4)

In terms of the $u$ and $v$ defined for Modular Equations,

$$M_3 = {v\over v+2u^3}={2v^3-u\over 3u}$$ (5)

$$M_5 = {v(1-uv^3)\over v-u^5}={u+v^5\over 5u(1+u^3v)}$$ (6)

$$M_7 = {v(1-uv)[1-uv+(uv)^2]\over v-u^7}$$

$$= {v^7-u\over 7u(1-uv)[1-uv+(uv)^2]}.$$ (7)