# Klein complete invariant

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## 6 matching pages

##### 1: 23.19 Interrelations
23.19.2 $J\left(\tau\right)=\frac{4}{27}\frac{\left(1-\lambda\left(\tau\right)+{\lambda% }^{2}\left(\tau\right)\right)^{3}}{\left(\lambda\left(\tau\right)\left(1-% \lambda\left(\tau\right)\right)\right)^{2}},$
23.19.3 $J\left(\tau\right)=\frac{{g_{2}}^{3}}{{g_{2}}^{3}-27{g_{3}}^{2}},$
##### 2: 23.15 Definitions
###### Klein’s CompleteInvariant
23.15.7 $J\left(\tau\right)=\frac{\left({\theta_{2}}^{8}\left(0,q\right)+{\theta_{3}}^{% 8}\left(0,q\right)+{\theta_{4}}^{8}\left(0,q\right)\right)^{3}}{54\left(\theta% _{1}'\left(0,q\right)\right)^{8}},$
##### 3: 23.1 Special Notation
The main functions treated in this chapter are the Weierstrass $\wp$-function $\wp\left(z\right)=\wp\left(z|\mathbb{L}\right)=\wp\left(z;g_{2},g_{3}\right)$; the Weierstrass zeta function $\zeta\left(z\right)=\zeta\left(z|\mathbb{L}\right)=\zeta\left(z;g_{2},g_{3}\right)$; the Weierstrass sigma function $\sigma\left(z\right)=\sigma\left(z|\mathbb{L}\right)=\sigma\left(z;g_{2},g_{3}\right)$; the elliptic modular function $\lambda\left(\tau\right)$; Klein’s complete invariant $J\left(\tau\right)$; Dedekind’s eta function $\eta\left(\tau\right)$. …
##### 4: 23.16 Graphics
See Figures 23.16.123.16.3 for the modular functions $\lambda$, $J$, and $\eta$. …
##### 5: 23.17 Elementary Properties
23.17.5 $1728J\left(\tau\right)=q^{-2}+744+1\;96884q^{2}+214\;93760q^{4}+\cdots,$