# §23.19 Interrelations

 23.19.1 $\mathop{\lambda\/}\nolimits\!\left(\tau\right)=16\left(\frac{{\mathop{\eta\/}% \nolimits^{2}}\!\left(2\tau\right)\mathop{\eta\/}\nolimits\!\left(\tfrac{1}{2}% \tau\right)}{{\mathop{\eta\/}\nolimits^{3}}\!\left(\tau\right)}\right)^{8},$
 23.19.2 $\mathop{J\/}\nolimits\!\left(\tau\right)=\frac{4}{27}\frac{\left(1-\mathop{% \lambda\/}\nolimits\!\left(\tau\right)+{\mathop{\lambda\/}\nolimits^{2}}\!% \left(\tau\right)\right)^{3}}{\left(\mathop{\lambda\/}\nolimits\!\left(\tau% \right)\left(1-\mathop{\lambda\/}\nolimits\!\left(\tau\right)\right)\right)^{2% }},$
 23.19.3 $\mathop{J\/}\nolimits\!\left(\tau\right)=\frac{g_{2}^{3}}{g_{2}^{3}-27g_{3}^{2% }},$

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 $\Delta=(2\pi)^{12}{\mathop{\eta\/}\nolimits^{24}}\!\left(\tau\right).$ Symbols: $\mathop{\eta\/}\nolimits\!\left(\NVar{\tau}\right)$: Dedekind’s eta function (or Dedekind modular function), $\Delta$: discriminant and $\tau$: complex variable Referenced by: §23.19 Permalink: http://dlmf.nist.gov/23.19.E4 Encodings: TeX, pMML, png See also: info for 23.19