  We first find a sufficient condition for a product of theta constants to be a
Siegel modular function of a given even level. And, when $K_{(2p)}$ denotes the
ray class field of $K=\mathbb{Q}(e^{2\pi i/5})$ modulo $2p$ for an odd prime
$p$, we describe a subfield of $K_{(2p)}$ generated by the special value of
certain theta constant by using Shimura's reciprocity law.
