Let
(
X
,
x
)
(X,x)
be a numerically Gorenstein elliptic singularity with
C
∗
{\mathbb {C}^{\ast }}
-action and
π
:
(
X
~
,
A
)
→
(
X
,
x
)
\pi :(\tilde X,A) \to (X,x)
the minimal good resolution. Assume that the central curve of
π
\pi
is an elliptic curve. We will determine the configuration of the w.d. graph (weighted dual graph) of
A
A
and obtain a condition for
(
X
,
x
)
(X,x)
to be a maximally elliptic singularity in the sense of Yau.