I am given an elliptic curve with homogeneous equation $X^3+Y^3+Z^3=0$ over a field K and I am asked to find the Weierstrass normal form.
I started by noticing that the point $(1,-1,0)$ is in the curve. Then, I find the tangent line at the point: $$\frac{\partial F}{\partial X}(X-1)+\frac{\partial F}{\partial Y}(Y+1)+\frac{\partial F}{\partial Z}(Z-0) = 0$$ to get $$Y=-X$$ But when I plug this in the initial equation I get $$Z^3 = 0$$ and here I stopped the process because these seem strange.
On
It is just $(1,-1,0)$ is an inflection point. So I believe the algorithm you were using initially does not work in this case.
The algorithm when your point is an inflection point is as follows:
1) Take tangent at point to be $z=0$
2)choose any line through point, call it $x=0$
3) choose line not through point, call it $y=0$
In step 1) use $z=X+Y$,
2) $Z=0$ passes through your point, so you may set $x=Z$
3) $X=0$ does not, so you may set $y=X$
If you do it correctly, one possible solution is $y^2=x^3-432$
HINT: Make a change of variables and use the identity $$X^3+Y^3=(X+Y)^3-3Y(X+Y)^2+3Y^2(X+Y)$$