sage: R.<u,v,a2,a4,u4,v4>=QQ[]
sage: S=R.quotient([
....:   v^2-u^3-a2*u^2-a4*u
....: , a4-u4^2
....: , numerator(a2-(v4/u4)^2+2*u4)
....: ])
sage: d=1-4*u4^3/v4^2
sage: x=(v4*u)/(u4*v)
sage: y=(u-u4)/(u+u4)
sage: oncurve=x^2+y^2-1-d*x^2*y^2
sage: oncurve
(4*u^4*u4^2 - 8*u^3*u4^3 + 4*u^2*u4^4 - 4*u*v^2*u4^2 + 4*u^3*v4^2)/(u^2*v^2*u4 + 2*u*v^2*u4^2 + v^2*u4^3)
sage: numerator(oncurve)
4*u^4*u4^2 - 8*u^3*u4^3 + 4*u^2*u4^4 - 4*u*v^2*u4^2 + 4*u^3*v4^2
sage: S(numerator(oncurve))
0
sage: (1+y)/(1-y)
2*u/(2*u4)
sage: u4*(1+y)/(1-y)
u
sage: v4*u/(u4*x)
v
sage: