Bringing an Elliptic curve in homogenous form to Weierstrass form

Question

I have a family of curves given by $F(U,V,W)= U^3 +V^3 + W^3- 3\lambda UVW$ in $\mathbb{P^2C}$, with an origin $O = [1,-1,0]$. I am struggling to bring this to the form $y^2 = x^3 - ax +b$ as I can't seem to change the basis correctly.

I have the fact that $O$ is an inflection point, and that for $P = (X,Y,1)$ we have $-P = (Y,X,1)$.

Answer 1

Magma can do this. The result is complicated. Note that it's a composition of two maps, and the result is still not even in short Weierstrass form, it's in long Weierstrass form (but standard techniques can be used from here to bring it to short Weierstrass form).

Magma V2.25-6     Thu Mar 17 2022 08:35:16 on t460s    [Seed = 1259995709]
Type ? for help.  Type <Ctrl>-D to quit.
> R<l> := FunctionField(Rationals());  
> P2<U,V,W> := ProjectiveSpace(R,2);
> C := Curve(P2, U^3 + V^3 + W^3 - 3 * l * U * V * W);
> pt := C![1,-1,0];   
> EllipticCurve(C,pt);
Elliptic Curve defined by y^2 + 3*l^4/(l^3 - 1)*x*y = x^3 - 9*l^5/(l^6 - 2*l^3 +
    1)*x^2 - 27*l^10/(l^9 - 3*l^6 + 3*l^3 - 1)*x - 27*l^15/(l^12 - 4*l^9 + 6*l^6
    - 4*l^3 + 1) over Univariate rational function field over Rational Field
Mapping from: CrvPln: C to Elliptic Curve defined by y^2 + 3*l^4/(l^3 - 1)*x*y =
x^3 - 9*l^5/(l^6 - 2*l^3 + 1)*x^2 - 27*l^10/(l^9 - 3*l^6 + 3*l^3 - 1)*x - 
    27*l^15/(l^12 - 4*l^9 + 6*l^6 - 4*l^3 + 1) over Univariate rational function
field over Rational Field
Composition of Mapping from: CrvPln: C to Curve over Univariate rational 
function field over Rational Field defined by
U^3 + 3*l/(l^3 - 1)*U^2*W + 3*l^4/(l^3 - 1)*U*V*W + 3*l^4/(l^3 - 1)*V^2*W - 
    3*l^2/(l^3 - 1)*U*W^2 + l^3/(l^3 - 1)*W^3
with equations : 
U + V
-U
1/l*U + 1/l*V + W and
Mapping from: Curve over Univariate rational function field over Rational Field 
defined by
U^3 + 3*l/(l^3 - 1)*U^2*W + 3*l^4/(l^3 - 1)*U*V*W + 3*l^4/(l^3 - 1)*V^2*W - 
    3*l^2/(l^3 - 1)*U*W^2 + l^3/(l^3 - 1)*W^3 to Elliptic Curve defined by y^2 +
3*l^4/(l^3 - 1)*x*y = x^3 - 9*l^5/(l^6 - 2*l^3 + 1)*x^2 - 27*l^10/(l^9 - 3*l^6 +
    3*l^3 - 1)*x - 27*l^15/(l^12 - 4*l^9 + 6*l^6 - 4*l^3 + 1) over Univariate 
rational function field over Rational Field
with equations : 
-3*l^4/(l^3 - 1)*U
-9*l^8/(l^6 - 2*l^3 + 1)*V
W
and inverse
3*l^4/(l^3 - 1)*$.1
$.2
-9*l^8/(l^6 - 2*l^3 + 1)*$.3
>