# How is curve C(b) converted to elliptic curve?

How can we transform curve $$C(b)\ :\ (3+b)X^2Y + (9+b)XY^2 - (4+b)X^2Z + (3-b^2)XYZ + (-4+b)Y^2Z + (-9+b)XZ^2+bYZ^2$$ into an elliptic curve without valuing $b$?

Edited: Expression in code format:

(3+b)*X^2*Y + (9+b)*X*Y^2 - (4+b)*X^2*Z + (3-b^2)*X*Y*Z + (-4+b)*Y^2*Z + (-9+b)*X*Z^2 + b*Y*Z^2

edit retag close merge delete

Sort by » oldest newest most voted

I tried:

F.<b> = PolynomialRing(QQ)
R.<X,Y,Z> = PolynomialRing(F)
cubic = (3+b)*X^2*Y + (9+b)*X*Y^2 - (4+b)*X^2*Z + (3-b^2)*X*Y*Z + (-4+b)*Y^2*Z + (-9+b)*X*Z^2 + b*Y*Z^2
# fX, fY, fZ = WeierstrassForm(cubic, transformation=True)

WeierstrassForm(cubic)


This gives:

(-1/48*b^8 + 5/12*b^6 + 1/2*b^5 - 619/24*b^4 - 49/2*b^3 + 2755/12*b^2 - 2*b - 142345/48,
1/864*b^12 - 5/144*b^10 - 1/24*b^9 + 223/96*b^8 + 59/24*b^7 - 2095/54*b^6 - 983/24*b^5 \
+ 27441/32*b^4 + 19765/24*b^3 - 863495/144*b^2 + 2213/6*b + 50223077/864)


(Result was manually rearranged.)

more

Thank you very much for your answer, but it gives an error after the "WeierstrassForm(cubic)" command, it does not calculate. I do not understand why it gives an error.