**What is the geometric meaning of $\omega=dx/(2y+a_1x+a_3)$ for an elliptic curve?**

This question is an adjunct to MO Q1 on formal laws and L-series, which motivated Q2. Q1 (Silverman) and Darmon (pg. 6) state:

The invariant holomorphic differential form (Neron differential) attached to an elliptic curve is

$\omega=dx/(2y+a_1x+a_3)$.

(Ancilliary question: Relation to Weierstrass's elliptic functions?)

I'd like to broaden the question as a community wiki to ask, "What are some interesting manifestations of this one-form in various families of elliptic curves?"

E.g., J. Hoffman in Topics in Elliptic Curves and Modular Forms gives for the Jacobi quartic family of elliptic curves

$\omega=dx/(1+2\kappa x^{2}+x^{4})^{1/2}=\sum_{n=0}^{\infty}L_{n}(\kappa)x^{2n}dx$

with $L_{n}(\kappa)$ the Legendre polynomials.