Solution
= Solution
Use the alternative affine coordinates
$$
t=-\frac{x}{y},\qquad w=-\frac1y,
$$
so that $x=t/w$ and $y=-1/w$. The equation becomes
$$
w=t^3+atw^2+bw^3.
$$
Recursive coefficient comparison gives a unique series $w(t)\in t^3k[[t]]$. Since $v(w(t))=3v(t)$ and hence $v(x)=-2v(t)$, $v(y)=-3v(t)$, this gives the <formal coordinates on a short Weierstrass curve> identification
$$
E_1(K)=\{(t,w(t)):v(t)\geq1\}.
$$