# ag.algebraic geometry – Log canonical surface with an elliptic singularity

I would like to know if there is an example as follows:

$$X$$ is a log canonical surface and $$x in X$$ is an elliptic singularity such that

1. The minimal resolution of $$x$$ is a circle of rational curves (or a single nodal rational curve).
2. The singularity $$x$$ is non-$$mathbb{Q}$$-factorial.

I think it looks reasonable, but I do not know any explicit example of such a surface. I searched some papers but none of them discuss the $$mathbb{Q}$$-factoriality.