I want to convert the cubic $x^3 + 3y^3 - 11z^3 = 0$ to Weierstrass form (to find its rank) so I tried to follow the suggestion from Timothy: I found three points $(2:1:1),(28:-19:5),(-537656:443213:212645)$ on the curve and used a linear change of variables to map them to $(1:0:0),(0:1:0),(0:0:1)$ then I dehomogenized and set y' = xy then homogenized that to get 1330 x'^3 - 252013629 x'^2 z' = y'^2 z' - 2352637000 y' z'^2 - 56454 x' y' z'. It doesn't seem to be birationally equivalent to what I started with (due to the rehomogenization), $(1:0:0)$ isn't a zero anymore (but $(1:0:9480240/201019314751)$ is).
So if anyone could tell me where I went wrong, or suggest a different technique or reference on how to convert elliptic curves into Weierstrass form that would be a big help. Thanks.