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 KleinInvariantJ KleinInvariantJ[] gives the Klein invariant modular elliptic function . Mathematical function (see Section A.3.10). The argument is the ratio of Weierstrass half-periods . KleinInvariantJ is given in terms of Weierstrass invariants by . is invariant under any combination of the modular transformations and . See SectionÂ 3.2.11 for a discussion of argument conventions for elliptic functions. See the Mathematica book: Section 3.2.11. See also: ModularLambda, DedekindEta, WeierstrassInvariants, EllipticTheta. Further Examples 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. In[1]:= Out[1]= 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). In[2]:= Out[2]=