Consider the elliptic curve defined by the Weierstrass equation $$E:y^2 = x^3+2$$ This defines a minimal Weierstrass equation for $p=2$ over $\mathbb{Q}_{2}$. It moreover has bad reduction at $p=2$. For the invariant differential on an elliptic curve, we know that it is holomorphic and nonvanishing. But what happens when considering the differential $\frac{dx}{2y}$ modulo $2$?
Is it in general possible that the associated invariant differential to a minimal Weierstrass equation for a prime $p$ vanishes modulo $p$?