This is documentation for Mathematica 3, which was
based on an earlier version of the Wolfram Language.
View current documentation (Version 11.1)
 Documentation / Mathematica / Built-in Functions / New in Version 3.0 / Mathematical Functions  /
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 Section 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]=