# §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 23
 23.19.3 $J\left(\tau\right)=\frac{g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2}},$ ⓘ Symbols: $J\left(\NVar{\tau}\right)$: Klein’s complete invariant, $g_{2}$, $g_{3}$: lattice invariants and $\tau$: complex variable Permalink: http://dlmf.nist.gov/23.19.E3 Encodings: TeX, pMML, png See also: Annotations for 23.19 and 23

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 $\Delta=(2\pi)^{12}{\eta^{24}}\left(\tau\right).$ ⓘ Symbols: $\eta\left(\NVar{\tau}\right)$: Dedekind’s eta function (or Dedekind modular function), $\pi$: the ratio of the circumference of a circle to its diameter, $\Delta$: discriminant and $\tau$: complex variable Referenced by: §23.19 Permalink: http://dlmf.nist.gov/23.19.E4 Encodings: TeX, pMML, png See also: Annotations for 23.19 and 23