Re: question on plane cubic curves <-> Weierstrass elliptic curve maps

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Apr 08, 2025 at 04:44:49PM -0700, American Citizen wrote:
> My question is this, can we find simpler maps for the (1) -> (2) and (2) ->
> (1) mappings?

Of course not, the maps and the curve are linked.
You can either have simple curves or simple maps, not both.

But the formula is not that bad in Jacobian coordinates.
[E,P]=ellfromeqn(2*a*b*y*(x+1)*(x-1) - (a+b)*(a-b)*x*(y+1)*(y-1),[x,y,1])

? P
%175 = [(-4*b^2*a^6+8*b^4*a^4-4*b^6*a^2)*x^4+(2*b*a^7+2*b^3*a^5-2*b^5*a^3-2*b^7*a)*y*x^3+((-a^8+2*b^4*a^4-b^8)*y^2+(a^8-8*b^2*a^6+14*b^4*a^4-8*b^6*a^2+b^8))*x^2+(-2*b*a^7-2*b^3*a^5+2*b^5*a^3+2*b^7*a)*y*x+(-4*b^2*a^6+8*b^4*a^4-4*b^6*a^2),(-8*b^2*a^10+64*b^4*a^8-112*b^6*a^6+64*b^8*a^4-8*b^10*a^2)*x^5+(8*b^2*a^10-64*b^4*a^8+112*b^6*a^6-64*b^8*a^4+8*b^10*a^2)*x,2*b*a*x^2+(-2*a^2+2*b^2)*y*x-2*b*a]

For (2) -> (1), I did not do the computation.

Cheers,
Bill.