# §23.19 Interrelations

 23.19.1 $\lambda\left(\tau\right)=16\left(\frac{{\eta^{2}}\left(2\tau\right)\eta\left(% \tfrac{1}{2}\tau\right)}{{\eta^{3}}\left(\tau\right)}\right)^{8},$
 23.19.2 $J\left(\tau\right)=\frac{4}{27}\frac{\left(1-\lambda\left(\tau\right)+{\lambda% ^{2}}\left(\tau\right)\right)^{3}}{\left(\lambda\left(\tau\right)\left(1-% \lambda\left(\tau\right)\right)\right)^{2}},$ ⓘ Symbols: $J\left(\NVar{\tau}\right)$: Klein’s complete invariant, $\lambda\left(\NVar{\tau}\right)$: elliptic modular function and $\tau$: complex variable Permalink: http://dlmf.nist.gov/23.19.E2 Encodings: TeX, pMML, png See also: Annotations for §23.19 and Ch.23
 23.19.3 $J\left(\tau\right)=\frac{{g_{2}^{3}}}{{g_{2}^{3}}-27{g_{3}^{2}}},$

where $g_{2},g_{3}$ are the invariants of the lattice $\mathbb{L}$ with generators $1$ and $\tau$; see §23.3(i).

Also, with $\Delta$ defined as in (23.3.4),

 23.19.4 $\Delta=(2\pi)^{12}{\eta^{24}}\left(\tau\right).$ ⓘ