The Dedekind eta function is defined by , for in the complex upper half-plane. If is regarded as the ratio of half-periods of an elliptic curve and are the corresponding Weierstrass coefficients, then .
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Under fractional linear transformations of the complex plane with integer coefficients, the Dedekind eta function transforms in simple ways. The most obvious example of such a transformation (apart from the identity) is ; in this case .
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] gives the Dedekind eta modular elliptic function 
. 
plane. It is not defined for real 
. 
is the ratio of Weierstrass half-periods 
. 
where 
is the discriminant, given in terms of Weierstrass invariants by 
. 
is defined by 
, for 
in the complex upper half-plane. If 
is regarded as the ratio of half-periods of an elliptic curve and 
are the corresponding Weierstrass coefficients, then 
. 
; in this case 
. 
