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Defining Hessian Curve

asked 2020-08-17 06:05:26 -0500

kelalaka gravatar image

How do we define a Hessian Curve on SageMath. I've failed to find a source about this. The curve equation is this

$$X^3 + Y^3 + cZ^3 = dXYZ$$ and paremetrazied with $c$ and $d$ over the field $GF(p)$

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answered 2020-08-18 07:19:04 -0500

FrédéricC gravatar image

updated 2020-08-19 02:54:01 -0500

Like this

sage: X,Y,Z=polygens(GF(7),'X,Y,Z')                                             
sage: C = Curve(X**3+Y**3+4*Z**3-2*X*Y*Z)                                       
sage: C.genus()                                                                 
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EDIT:

sage: EllipticCurve_from_cubic(X**3+Y**3+4*Z**3-2*X*Y*Z)                        
Scheme morphism:
  From: Projective Plane Curve over Finite Field of size 7 defined by X^3 + Y^3 - 2*X*Y*Z - 3*Z^3
  To:   Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 6*x^2 + 4*x + 4 over Finite Field of size 7
  Defn: Defined on coordinates by sending (X : Y : Z) to
        (-Z : X : 2*X + Y - 2*Z)
sage: _.codomain()                                                              
Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 6*x^2 + 4*x + 4 over Finite Field of size 7
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OK, This defines a general curve. Now it is hard to use the Elliptic curves over finite fields tools. Is there a way?

kelalaka gravatar imagekelalaka ( 2020-08-18 14:50:50 -0500 )edit

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Asked: 2020-08-17 06:05:26 -0500

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Last updated: Aug 19