<s>[INST] Birationally transforming general curve of genus 1 to Weierstrass form 
What are general rules to birationally transform general curve of any
  degree of genus 1 to Weierstrass form, provided we have one rational
  point?

Example of curve of degree 12:
$$x^9 y^3+9 x^9 y^2+27 x^9 y+27 x^9+9 x^8 y^3+81 x^8 y^2+243 x^8 y+243 x^8+35 x^7 y^3+318 x^7 y^2+963 x^7 y+972 x^7+74 x^6 y^3+687 x^6 y^2+2124 x^6 y+2187 x^6+90 x^5 y^3+871 x^5 y^2+2799 x^5 y+2988 x^5+67 x^4 y^3+692 x^4 y^2+2358 x^4 y+2655 x^4+39 x^3 y^3+415 x^3 y^2+1466 x^3 y+1717 x^3+21 x^2 y^3+211 x^2 y^2+723 x^2 y+840 x^2+4 x y^3+47 x y^2+180 x y+228 x-3 y^3-20 y^2-40 y-20=0$$
This curve has (geometric) genus 1. It has rational point $(-3, -\frac{17}{5})$.
I am familiar with transforming $y^2=a x^4+b x^3+c x^2+d x+e$ to Weierstrass form but have never seen similar process for curves of higher degree than $4$.
 [/INST] $$C: x^9 y^3+9 x^9 y^2+27 x^9 y+27 x^9+9 x^8 y^3+81 x^8 y^2+243 x^8 y+243 x^8+35 x^7 y^3+318 x^7 y^2+963 x^7 y+972 x^7+74 x^6 y^3+687 x^6 y^2+2124 x^6 y+2187 x^6+90 x^5 y^3+871 x^5 y^2+2799 x^5 y+2988 x^5+67 x^4 y^3+692 x^4 y^2+2358 x^4 y+2655 x^4+39 x^3 y^3+415 x^3 y^2+1466 x^3 y+1717 x^3+21 x^2 y^3+211 x^2 y^2+723 x^2 y+840 x^2+4 x y^3+47 x y^2+180 x y+228 x-3 y^3-20 y^2-40 y-20=0$$
$$E: v^2=u^3+2 u^2+3 u+7$$
$$C\to E: \{x,y\}=\left\{\frac{2-u-v}{u+v},\frac{(3 u-3 v+2) (u+v)^2+24}{(v-u) (u+v)^2-8}\right\}$$
$$E\to C: \{u,v\}=\left\{\frac{-x^3 y-3 x^3-3 x^2 y-9 x^2-3 x y-10 x-1}{(x+1) (y+3)},\frac{x^3 y+3 x^3+3 x^2 y+9 x^2+3 x y+10 x+2 y+7}{(x+1) (y+3)}\right\}$$
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