  We study some 2 and 3-isogenies on the singular $K3$ surface $Y_{10}$ of
discriminant 72 and belonging to the Ap\'ery-Fermi pencil $(Y_k)$, and find on
it many interesting properties. For example some of its elliptic fibrations
with 3-torsion section induce by 3-isogeny either an elliptic fibration of
$Y_2$, the unique $K3$ surface of discriminant 8, or an elliptic fibration of
other $K3$ surfaces of discriminant 72.
