Let $K$ be a local field and $E/K$ an elliptic curve such that the set of $N$-torsion points, $E[N]$, is contained in $E(K)$. For $e$ in $E[N]$, I am interested in finding and expression for the function $f_e$ in K(E) such that $div(f_e)=N(e)-N(0)$. More concretely, I am interested in the values of $f_e$ when restricted to the formal group of $E$. I guess using the Weierstrass preparation theorem could gives us an idea of how $f_e$ can be factored, but what about the coefficients of the series that appear on the product?
I know there is an algorithm, Miller's algorithm, that allows one to compute f_e, but the expression is quite complicate. There is also a $p$-adic analogue of the sigma function, due to Tate, for the case in which E has ordinary reduction, which allows one to find an expression for f_e as a product of these functions. What about the case of super-singular reduction? the general case?
Any insights or references would be greatly appreciated.