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 $$\omega = \frac{dx}{2y+a_1x+a_3},$$ 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*. Here is claimed that any two invariant differentials on the elliptic curve $E_\Lambda$ are scalar multiples of one another. But strictly speaking we have defined only one invariant differential on $E_\Lambda$

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