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Klein complete invariant

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1: 23.19 Interrelations
23.19.2 J ( τ ) = 4 27 ( 1 - λ ( τ ) + λ 2 ( τ ) ) 3 ( λ ( τ ) ( 1 - λ ( τ ) ) ) 2 ,
23.19.3 J ( τ ) = g 2 3 g 2 3 - 27 g 3 2 ,
2: 23.15 Definitions
Klein’s Complete Invariant
23.15.7 J ( τ ) = ( θ 2 8 ( 0 , q ) + θ 3 8 ( 0 , q ) + θ 4 8 ( 0 , q ) ) 3 54 ( θ 1 ( 0 , q ) ) 8 ,
3: 23.1 Special Notation
The main functions treated in this chapter are the Weierstrass -function ( z ) = ( z | 𝕃 ) = ( z ; g 2 , g 3 ) ; the Weierstrass zeta function ζ ( z ) = ζ ( z | 𝕃 ) = ζ ( z ; g 2 , g 3 ) ; the Weierstrass sigma function σ ( z ) = σ ( z | 𝕃 ) = σ ( z ; g 2 , g 3 ) ; the elliptic modular function λ ( τ ) ; Klein’s complete invariant J ( τ ) ; Dedekind’s eta function η ( τ ) . …
4: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions λ , J , and η . …
See accompanying text
Figure 23.16.1: Modular functions λ ( i y ) , J ( i y ) , η ( i y ) for 0 y 3 . … Magnify
5: 23.18 Modular Transformations
Klein’s Complete Invariant
23.18.4 J ( 𝒜 τ ) = J ( τ ) .
6: 23.17 Elementary Properties
23.17.5 1728 J ( τ ) = q - 2 + 744 + 1 96884 q 2 + 214 93760 q 4 + ,