The Klein invariant is an invariant of elliptic curves, equal to , where are the Weierstrass coefficients. The function KleinInvariantJ takes as its argument the ratio between half-periods of the curve, rather than the Weierstrass coefficients.
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Two ratios and yield the same if and only if , for some integers with . Values of thus related can be regarded as coming from different possible pairs of periods for the same elliptic curve (that is, different fundamental parallelograms).
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] gives the Klein invariant modular elliptic function 
. 
is the ratio of Weierstrass half-periods 
. 
. 
is invariant under any combination of the modular transformations 
and 
. 
is an invariant of elliptic curves, equal to 
, where 
are the Weierstrass coefficients. The function KleinInvariantJ takes as its argument the ratio between half-periods of the curve, rather than the Weierstrass coefficients. 
and 
yield the same 
if and only if 
, for some integers 
with 
. Values of 
thus related can be regarded as coming from different possible pairs of periods for the same elliptic curve (that is, different fundamental parallelograms). 
