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

class sage.schemes.plane_quartics.quartic_generic.QuarticCurve_generic(A, f)

Bases: sage.schemes.curves.projective_curve.ProjectivePlaneCurve

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

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