Let $E/L$ be an elliptic curve defined over a number field $L$. Assume moreover that $E$ has complex multiplication by an imaginary quadratic field $K$. Let $\mathfrak{p}$ be a prime ideal of $K$. Assume that $E[\mathfrak{p}]\subseteq E(L)$ and that $E$ has good reduction at $\wp$ where $\wp$ is a prime ideal of $L$ above $\mathfrak{p}\cap \mathbb{Z}$. Consider the reduction map $$ \pi: E(L)\rightarrow \tilde{E}(\mathcal{O}_L/\wp), $$ where $\tilde{E}$ denotes the reduction of $E$ modulo $\wp$.

Consider the following two statements:

S1: $\ker(\pi)\supseteq E[\mathfrak{p}]$

S2: $\tilde{E}[\mathfrak{p}](\mathcal{O}_L/\wp)=\{0\}$

Note that S2 implies S1.

Q: Is S2 true, if so, how does one prove it ?

This does not seem to follow directly from the theory of formal groups.