Birationally transforming general curve of genus 1 to Weierstrass form

Question

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 $4$.

Answer 1

$$C: x^9 y^3+9 x^9 y^2+27 x^9 y+27 x^9+9 x^8 y^3+81 x^8 y^2+243 x^8 y+243 x^8+35 x^7 y^3+318 x^7 y^2+963 x^7 y+972 x^7+74 x^6 y^3+687 x^6 y^2+2124 x^6 y+2187 x^6+90 x^5 y^3+871 x^5 y^2+2799 x^5 y+2988 x^5+67 x^4 y^3+692 x^4 y^2+2358 x^4 y+2655 x^4+39 x^3 y^3+415 x^3 y^2+1466 x^3 y+1717 x^3+21 x^2 y^3+211 x^2 y^2+723 x^2 y+840 x^2+4 x y^3+47 x y^2+180 x y+228 x-3 y^3-20 y^2-40 y-20=0$$

$$E: v^2=u^3+2 u^2+3 u+7$$

$$C\to E: \{x,y\}=\left\{\frac{2-u-v}{u+v},\frac{(3 u-3 v+2) (u+v)^2+24}{(v-u) (u+v)^2-8}\right\}$$ $$E\to C: \{u,v\}=\left\{\frac{-x^3 y-3 x^3-3 x^2 y-9 x^2-3 x y-10 x-1}{(x+1) (y+3)},\frac{x^3 y+3 x^3+3 x^2 y+9 x^2+3 x y+10 x+2 y+7}{(x+1) (y+3)}\right\}$$

Answer 2

In case someone would be interested in Maple's result obtained by user Jan-Magnus Økland, I rewrote it by hand and simplified a bit:

x0^3+(4096/375)*x0-39583744/421875+y0^2=0

{x0=(-64*(459-2883*x-9835*x^2-10922*x^3+911*x^4+32019*x^5+79032*x^6+
107703*x^7+90099*x^8+47466*x^9+15525*x^10+2916*x^11+243*x^12+477*y-
1650*x*y-5739*x^2*y-4926*x^3*y+2229*x^4*y+18462*x^5*y+46482*x^6*y+
66672*x^7*y+57789*x^8*y+31104*x^9*y+10296*x^10*y+1944*x^11*y+162*x^12*y+
108*y^2-243*x*y^2-867*x^2*y^2-468*x^3*y^2+684*x^4*y^2+2697*x^5*y^2+
6786*x^6*y^2+10281*x^7*y^2+9255*x^8*y^2+5094*x^9*y^2+1707*x^10*y^2+
324*x^11*y^2+27*x^12*y^2))/(75*(1+x)^2*(3+x)^2),
y0=(512*(843-2710*x-16519*x^2-12074*x^3+53732*x^4+202061*x^5+420043*x^6+
605147*x^7+603976*x^8+410519*x^9+186465*x^10+54489*x^11+9342*x^12+
720*x^13+663*y-1555*x*y-10411*x^2*y-5199*x^3*y+37741*x^4*y+122433*x^5*y+
248798*x^6*y+368668*x^7*y+380591*x^8*y+265351*x^9*y+122554*x^10*y+
36166*x^11*y+6228*x^12*y+480*x^13*y+117*y^2-252*x*y^2-1708*x^2*y^2-
447*x^3*y^2+6786*x^4*y^2+18853*x^5*y^2+36789*x^6*y^2+55939*x^7*y^2+
59835*x^8*y^2+42846*x^9*y^2+20133*x^10*y^2+6001*x^11*y^2+1038*x^12*y^2+
80*x^13*y^2))/(125*(1+x)^2*(3+x)^3)}

{x=(-3*(641990656-26112000*x0+23400000*x0^2+421875*x0^3-18432000*y0+
4050000*x0*y0))/(1493958656+4608000*x0+81000000*x0^2+421875*x0^3),
y=-((12471594100595144366690926592+1955419142592015308213452800*x0+
1878075405942780183183360000*x0^2+317367943168385875968000000*x0^3+
100956115396303257600000000*x0^4+10300549541068800000000000*x0^5+
1553373542016000000000000*x0^6+92061983700000000000000*x0^7+
1575601083984375000000*x0^8+6382198333740234375*x0^9-
85483495844938839490560000*y0+18695958056411332608000000*x0*y0-
10870651215214018560000000*x0^2*y0+2004895847153664000000000*x0^3*y0-
253485556531200000000000*x0^4*y0+35822535120000000000000*x0^5*y0+
1409384812500000000000*x0^6*y0-33397668457031250000*x0^7*y0)/
(4064816181869106669044105216+616121378211946711272652800*x0+
627407206768386936668160000*x0^2+103220483647881609216000000*x0^3+
34217094666623385600000000*x0^4+3521701778227200000000000*x0^5+
545117693184000000000000*x0^6+34660450800000000000000*x0^7+
680639677734375000000*x0^8+1877117156982421875*x0^9))}