In our earlier paper it was proved that the singular locus of
A
g
A_{g}
(coarse moduli space of principally polarized abelian varieties over
C
\mathbb {C}
) is expressed as the union of irreducible varieties
A
g
(
p
,
α
)
A_{g}(p,\alpha )
representing abelian varieties with an order
p
p
automorphism of fixed entire representation. In this paper we prove that
A
g
(
p
,
α
)
A_{g}(p,\alpha )
is an irreducible component of
Sing
A
g
\text {Sing} A_{g}
if and only if for a general element of this variety its automorphism group modulo
{
±
1
}
\{\pm 1\}
,
G
+
G_{+}
, satisfies the equivalent conditions:
G
+
=
⟨
α
⟩
G_{+}=\langle \alpha \rangle
or
N
G
+
(
⟨
α
⟩
)
=
⟨
α
⟩
N_{G_{+}}(\langle \alpha \rangle )=\langle \alpha \rangle
. We illustrate how these results can be used by studying the case
g
=
4
g=4
and
p
=
5
p=5
.