Explicit $t$-expansions for the elliptic curve $y^{2}= 4(x^{3}+Ax+B)$

Abstract

For an elliptic curve $E: y^{2} = 4(x^{3} + A x +B)$ over a field of characteristic $\neq 2$, we explicitly compute the pullback to the formal completion of $E$ at the origin of some important objects on $E$ including the functions $x$, $y$ and the invariant differential $\omega=dx/y$ in terms of the formal parameter $t = -2x/y$.