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?)

When wading into Darmon's paper and Silverman's book

I'd like to broaden the question as a community wiki to ask, I'm quickly "What are some interesting manifestations of this one-form in way over my head, so I'm looking for an interpretation at the level various families of say Terry Tao's Diffelliptic curves?"

E.g., J. Forms Hoffman in Topics in Elliptic Curves and Integration, or the explanations often found Modular Forms gives for an autonomous differential equation the Jacobi quartic family of elliptic curves

$(dy-fdx)(V)=0$.\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)$the Legendre polynomials. 3 edited tags 2 deleted 7 characters in body What is the geometric meaning of$\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)$. (Ancilliary question: Relation to Weierstrass's elliptic functions?) When wading into Darmon's paper and Silverman's book, I'm quickly in way over my head, so I'm looking for an interpretation at the level of say Terry Tao's Diff. Forms and Integration, or the explanations often found for an autonomous differential equation$(dy-fdx)(V)=0\$.

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