  Special values of Siegel modular functions for $\operatorname{Sp}
(\mathbb{Z})$ generate class fields of CM fields. They also yield abelian
varieties with a known endomorphism ring. Smaller alternative values of modular
functions that lie in the same class fields (class invariants) thus help to
speed up the computation of those mathematical objects.
  We show that modular functions for the subgroup $\Gamma^0 (N)\subseteq
\operatorname{Sp}(\mathbb{Z})$ yield class invariants under some splitting
conditions on $N$, generalising results due to Schertz from classical modular
functions to Siegel modular functions. We show how to obtain all Galois
conjugates of a class invariant by evaluating the same modular function in CM
period matrices derived from an \emph{$N$-system}. Such a system consists of
quadratic polynomials with coefficients in the real-quadratic subfield
satisfying certain congruence conditions modulo $N$. We also examine conditions
under which the minimal polynomial of a class invariant is real.
  Examples show that we may obtain class invariants that are much smaller than
in previous constructions.
