Let $E$ be an elliptic curve over $\mathbb{Q}$. Let us fix a rational prime $\ell$ (odd, if you want) and an $\ell$-torsion point $P\in E[\ell]$. For a prime $\mathfrak{p}$ of $K=\mathbb{Q}(E[\ell])$, let us say that $P$ has *nonsingular reduction* at $\mathfrak{p}$, if $\tilde{P}\neq\tilde{O}$ in $\tilde{E}_\mathfrak{p}$ (the reduction of $E$ at $\mathfrak{p}$, so I implicitly assumed $E$ has good reduction at $p=\mathfrak{p}\cap\mathbb{Z}$), otherwise we say that $P$ has *singular reduction* at $\mathfrak{p}$. Let $S_P$ be the set of primes of singular reduction for $P$. With this setting, my question is

Can we have $S_P=\{\text{primes over }\ell\}$?