I am somewhat confused by the definition of the invariant differentials in J. Silverman's book The Arithmetic of Elliptic Curves.

Let $E$ be an elliptic curve with Weierstrass equation $F(x,y)=0$. Then the invariant differential corresponding to the Weierstrass equation is defined as $$\omega = \frac{dx}{2y+a_1x+a_2},$$ where the $a_i$ are as usual the coefficients of $F(x,y)$.

However, in formulation of various theorems, for example Theorem 5.2. on page 77, the notion of an invariant differential is used for a general elliptic curve, without any explicit reference to any particular Weierstrass equation.

Thus given an elliptic curve $E$, what is meant by an invariant differential on $E$?