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 explicit reference to any particular Weierstrass equation.

Another example is Proposition 1.1. in the book Advanced Topics in the Arithmetic of Elliptic curves.

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