Elliptic fibrations of $K3$ surfaces belonging to the Ap\'ery-Fermi pencil ($Y_k$) may have $2$ or $3$-torsion sections defining on $(Y_k)$ automorphisms $\tau$ of order $2$ or $3$. First we consider $Y_{ k}/\tau$ \ for some fibrations of the singular $K3$ surface $Y_{ 10}$ in the case of two-torsion sections and obtain as for the singular surface $Y_{ 2}$ either the Kummer surface associated to $Y_{ 10}$ or $Y_{ 10}$ itself. This last case is associated with the complex multiplication on $Y_{ 10}$. We prove also that for all the fibrations of $Y_{ 2}$ with $3$-torsion sections $Y_{ 2}/\tau=Y_{ 10}.$ Results are different for $Y_{ 10}$ where we can obtain for $Y_{ 10}/\tau$ one of the two surfaces with transcendental lattice $[4 \quad 0\quad 18]$ or $\left[2 \quad 0 \quad 36\right]$. We also explicitly link $3$-isogeny on a fibration and base change on other fibrations.
Categories: math.AG