  Using the theory of rational elliptic fibrations, we construct and discuss a
one parameter family of configurations of $12$ conics and $9$ points in the
projective plane that realizes an abstract configuration $(12_6,9_8)$. This is
analogous to the famous Hesse configuration of $12$ lines and $9$ points
forming an abstract configuration $(12_3,9_4)$. We also show that any Halphen
elliptic fibration of index $2$ with four triangular singular fibers arises
from such configuration of conics.
