For an elliptic curve $y^2=4x^3-g\_2x-g\_3$ over $\mathbb{Q}$, it is known that the local parameter at $O$ (the identity point) could be written by $-\frac{x}{y}$. Is it possible to write down the local parameter at $P\in E[\ell]$ for some prime $\ell\geq3$ with $P\neq O$ in the similar way (maybe, in terms of $x$ and $y$)?



Local parameter at torsion points of elliptic curve
Let $P=(a,b)$. Then $x-a$ is a local parameter at $P$, since it has a zero of order one at P. (It's two zeros are $P$ and $-P$.) Or if you want a local parameter that is defined over the field of definition of $E$, you can take the $\ell$-division polynomial $$\psi\_\ell(x):=\prod\_{P\in E[\ell]/\{\pm1\}} \bigl(x-x(P)\bigr).$$


But you may be misunderstanding what a local parameter is, because you say that $-x/y$ is *the* local parameter at $O$. It is just one of many local parameters. In general, a local parameter at a point $P$ on a smooth curve $C$ is simply a function $f$ in $K(C)$ with the property that $\text{ord}\_P(f)=1$, or equivalently, a function that generates the maximal ideal of the local ring at $P$. 


