Plane quartic curves over a general ring. These are generic genus 3 curves,¶
as distinct from hyperelliptic curves of genus 3.
EXAMPLES:
sage: PP.<X,Y,Z> = ProjectiveSpace(2, QQ)
sage: f = X^4 + Y^4 + Z^4 - 3*X*Y*Z*(X+Y+Z)
sage: C = QuarticCurve(f); C
Quartic Curve over Rational Field defined by X^4 + Y^4 - 3*X^2*Y*Z - 3*X*Y^2*Z - 3*X*Y*Z^2 + Z^4
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sage.schemes.plane_quartics.quartic_generic.QuarticCurve_generic(A, f)¶ Bases:
sage.schemes.curves.projective_curve.ProjectivePlaneCurveInitialization function.
EXAMPLES:
sage: P.<x,y,z> = ProjectiveSpace(QQbar, 2) sage: C = Curve([y*z - x^2 - QQbar.gen()*z^2], P); C Projective Plane Curve over Algebraic Field defined by -x^2 + y*z + (-I)*z^2
sage: P.<x,y,z> = ProjectiveSpace(GF(5^2, 'v'), 2) sage: C = Curve([y^2*z - x*z^2 - z^3], P); C Projective Plane Curve over Finite Field in v of size 5^2 defined by y^2*z - x*z^2 - z^3
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genus()¶ Returns the genus of self
EXAMPLES:
sage: x,y,z=PolynomialRing(QQ,['x','y','z']).gens() sage: Q = QuarticCurve(x**4+y**4+z**4) sage: Q.genus() 3
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sage.schemes.plane_quartics.quartic_generic.is_QuarticCurve(C)¶ Checks whether C is a Quartic Curve
EXAMPLES:
sage: from sage.schemes.plane_quartics.quartic_generic import is_QuarticCurve sage: x,y,z=PolynomialRing(QQ,['x','y','z']).gens() sage: Q = QuarticCurve(x**4+y**4+z**4) sage: is_QuarticCurve(Q) True