Plane quartic curves over a general ring - Plane and Space Curves

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# Plane quartic curves over a general ring

These are generic genus 3 curves, as distinct from hyperelliptic curves of genus 3.

EXAMPLES:

Sage

    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

Python

    >>> from sage.all import *
    >>> PP = ProjectiveSpace(Integer(2), QQ, names=('X', 'Y', 'Z',)); (X, Y, Z,) = PP._first_ngens(3)
    >>> f = X**Integer(4) + Y**Integer(4) + Z**Integer(4) - Integer(3)*X*Y*Z*(X+Y+Z)
    >>> 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, category=None)

  Bases: `ProjectivePlaneCurve`

  genus()

    Return the genus of `self`.

    EXAMPLES:

    Sage

        sage: x,y,z = PolynomialRing(QQ, ['x','y','z']).gens()
        sage: Q = QuarticCurve(x**4 + y**4 + z**4)
        sage: Q.genus()
        3

    Python

        >>> from sage.all import *
        >>> x,y,z = PolynomialRing(QQ, ['x','y','z']).gens()
        >>> Q = QuarticCurve(x**Integer(4) + y**Integer(4) + z**Integer(4))
        >>> Q.genus()
        3