# Defining Hessian Curve

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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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()
1

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
more

OK, This defines a general curve. Now it is hard to use the Elliptic curves over finite fields tools. Is there a way?

( 2020-08-18 21:50:50 +0200 )edit