  In this article we prove an analogue of a theorem of Lachaud, Ritzenthaler,
and Zykin, which allows us to connect invariants of binary oc-tics to Siegel
modular forms of genus 3. We use this connection to show that certain modular
functions, when restricted to the hyperelliptic locus, assume values whose
denominators are products of powers of primes of bad reduction for the
associated hyperelliptic curves. We illustrate our theorem with explicit
computations. This work is motivated by the study of the value of these modular
functions at CM points of the Siegel upper-half space, which, if their
denominators are known, can be used to effectively compute models of
(hyperelliptic, in our case) curves with CM.
