Cyclic torsion of elliptic curves

Let $E$ be an elliptic curve over a number field $k$ such that $\operatorname{End}_{k}E$
$= {\mathbf Z}$ and let $w(k)$ denote the number of roots of unity in $k$. Ross proposed a question: Is $E$ isogenous over $k$ to an elliptic curve $E'/k$ such that $E'(k)_{tors}$ is cyclic of order dividing $w(k)$? A counter-example of this question is given. We show that $E$ is isogenous to $E'/k$ such that $E'(k)_{tors}\,\subset{\mathbf Z}/w(k)^2{\mathbf Z}$. In case $E$ has complex multiplication and $\operatorname{End}_kE={\mathbf Z}$, we obtain certain criteria whether or not $E$ is isogenous to $E'/k$ such that $E'(k)_{tors}\,\subset{\mathbf Z}/2{\mathbf Z}$.