quartic_generic.py

"""
Plane quartic curves over a general ring. These are generic genus 3 curves,
as distinct from hyperelliptic curves of genus 3.

EXAMPLE:
    sage: PP, (X,Y,Z) = ProjectiveSpace(2,QQ,'X,Y,Z').objgens()
    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
"""

#*****************************************************************************
#  Copyright (C) 2006 David Kohel <[email protected]>
#  Distributed under the terms of the GNU General Public License (GPL)
#                  http://www.gnu.org/licenses/
#*****************************************************************************


import sage.schemes.plane_curves.projective_curve as projective_curve

def 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

    """
    return isinstance(C, QuarticCurve_generic)

class QuarticCurve_generic(projective_curve.ProjectiveCurve_generic):
    # DRK: Note that we should check whether the curve is

    def _repr_type(self):
        """
        Return the representation of self

        EXAMPLES::

            sage: x,y,z=PolynomialRing(QQ,['x','y','z']).gens()
            sage: Q = QuarticCurve(x**4+y**4+z**4)
            sage: Q._repr_type()
            'Quartic'
        """
        return "Quartic"

    def genus(self):
        """
        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
        """
        return 3