Birational transformation of general genus 1 curve to Weierstrass form

asked 2019-12-27 11:32:29 -0500

azerbajdzan gravatar image

Is it possible to birationally transform general curve of genus 1 of any degree to Weierstrass form using Sage?

Example of curve of degree $12$ which has genus $1$ and has rational point $(-3, -\frac{17}{5})$.

Input:

 x,y,z = QQ['x,y,z'].gens()
C = Curve(x^9*y^3 + 9*x^9*y^2*z + 9*x^8*y^3*z + 27*x^9*y*z^2 + 81*x^8*y^2*z^2 + 35*x^7*y^3*z^2 + 
 27*x^9*z^3 + 243*x^8*y*z^3 + 318*x^7*y^2*z^3 + 74*x^6*y^3*z^3 + 243*x^8*z^4 + 
 963*x^7*y*z^4 + 687*x^6*y^2*z^4 + 90*x^5*y^3*z^4 + 972*x^7*z^5 + 2124*x^6*y*z^5 + 
 871*x^5*y^2*z^5 + 67*x^4*y^3*z^5 + 2187*x^6*z^6 + 2799*x^5*y*z^6 + 692*x^4*y^2*z^6 + 
 39*x^3*y^3*z^6 + 2988*x^5*z^7 + 2358*x^4*y*z^7 + 415*x^3*y^2*z^7 + 21*x^2*y^3*z^7 + 
 2655*x^4*z^8 + 1466*x^3*y*z^8 + 211*x^2*y^2*z^8 + 4*x*y^3*z^8 + 1717*x^3*z^9 + 
 723*x^2*y*z^9 + 47*x*y^2*z^9 - 3*y^3*z^9 + 840*x^2*z^10 + 180*x*y*z^10 - 20*y^2*z^10 + 
 228*x*z^11 - 40*y*z^11 - 20*z^12)
C.genus()

Output:

1
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