Torsion points with exactly one singular prime (on elliptic curves)

Question

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 $S_P=\{\text{primes over }\ell\}$?

Answer 1

The status of your "implicit assumption" isn't quite clear -- do you want all primes where $E$ has bad reduction to be in $S_P$, or only a subset of them?

If you take E to be Cremona's 27a1, $y^2 + y = x^3 - 7$, then $E$ has a rational 3-torsion point $(3, -5)$, and for all rational primes $p \ne 3$ of E, the mod $p$ reduction of $P$ is automatically non-trivial. At $p = 3$, the reduction is bad additive and $P$ maps to the singular point of the reduced curve. Does this fit your requirements?

(What's making this work is that the kernel of reduction modulo $p$ cannot contain non-trivial torsion points of prime-to-$p$ order.)