Every elliptic curve over $\mathbb{Q}_p$ has a minimal integral Weierstrass model of the form $$ E: y^2 = x^3+Ax+B $$ with $A,B\in\mathbb{Z}_p$.
Here is a problem that I've run into. Suppose $p=3$ and define $E/\mathbb{Q}_3$ as above. I can reduce the coefficients of $E$ modulo $3$ in the naive way to get an elliptic curve $\overline{E}/\mathbb{F}_3$ as follows: $$ \overline{E}:y^2 =x^3+Ax+B \pmod 3 $$ For most pairs $A,B\in\mathbb{Z}_3$, such an $\overline{E}$ will be non-singular. However, Theorem 4.1 in Silverman's Arithmetic of Elliptic Curves tells me the following:
Let $E/\mathbb{F}_3$ be an elliptic curve given by a Weierstrass equation $$ E:y^2 = f(x) $$ where $f(x)\in \mathbb{F}_3[x]$ is a cubic with distinct roots in an algebraic closure. Then $E$ is supersingular if and only if the coefficient of $x^2$ in $f(x)$ is zero.
So by following my procedure as defined above, it seems like the reduction of every elliptic curve over $\mathbb{Q}_3$ will give me a supersingular elliptic curve. But this goes against my intuition that most elliptic curves over finite fields are ordinary (as the name suggests). Where did I go wrong?