Change of coordinates for $X^3+Y^3+60Z^3=0$

Question

I have to demonstrate that the curve $$ X^3+Y^3+60Z^3=0$$ is birationally equivalent to $$Y^2=X^3-2^43^360^2$$ or to $$Y^2=X^3-3^330^2.$$ I can't find a proper change of coordinates for this purpose. Does anyone know how can I do? Thanks!

Answer 1

It is known that the Selmer curve $$ X^3 + Y^3 = dZ^3 $$ is birationally equivalent to Weierstrass curve $$ V^2W = U^3 - 2^4 3^3 d^2 W^3 $$ under the mutually inverse transformations $X= V - 36dW, Y= -V-36dW , Z=-6U$ along with $ U=12dZ, V= -36d(X-Y), W=X+Y $. My source for this is here.

I think these will work with your case, i.e. $W=1$ and $d=-60$ though I'm not very well-versed in elliptic curves, so please correct me if this is wrong.