The parameter appearing in the definition of the incomplete elliptic integral is called the modulus. For fixed, can be regarded as a map from (most of) the complex plane into the elliptic curve with half-periods and . Thus from the modulus we obtain the invariant of the curve where the corresponding elliptic functions live. The modular function reverses this correspondence, that is, .
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The function is invariant under the group of transformations of the complex upper half-plane generated by and .
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] gives the modular lambda elliptic function 
. 
plane. It is not defined for real 
. 
is the ratio of Weierstrass half-periods 
. 
for elliptic functions in terms of 
according to 
. 
where the nome 
is given by 
. 
is invariant under any combination of the modular transformations 
and 
. 
appearing in the definition of the incomplete elliptic integral 
is called the modulus. For 
fixed, 
can be regarded as a map from (most of) the complex plane into the elliptic curve with half-periods 
and 
. Thus from the modulus we obtain the invariant 
of the curve where the corresponding elliptic functions live. 
reverses this correspondence, that is, 
. 
is invariant under the group of transformations of the complex upper half-plane generated by 
and 
. 
