Points are given by their projective coordinates, normalized depending on the base field.
Returns the image of the identity element on the Kummer surface K, which is normalized to be the origin (0 : 0 : 0 : 1).
Returns the point on the Kummer surface K defined by the projective coordinates x1, x2, x3, and x4.
Given a point P on the Jacobian of K, or on a Kummer surface for which K is a base extension, this returns the point on K.
Given a sequence S = [x1, x2, x3, x4] of elements of the base field of K, the function returns true if the point specified by the sequence defines the homogeneous coordinates of a point on the Kummer surface K. If so, the corresponding point on K is returned as the second value.
Returns the indexed set of points on the Kummer surface K with first three coordinates given by the sequence [x1, x2, x3].
Returns the i-th coordinate of the point P, for 1≤i≤4.
Given a point P on a Kummer surface, the function returns the coordinates of P as a sequence.
Given two points on the same Kummer surface, this returns true if and only if the points P and Q are equal.
Given two points on the same Kummer surface, this returns false if and only if the points P and Q are equal.
Returns the negation of the point P on the Kummer surface, equal to P itself.
Returns the n-th multiple of the point P on the Kummer surface K.
Returns the double 2 * P of the point P.
Let P and Q be points on the Jacobian J of a genus 2 curve. Given the images P1, P2, and P3 on the Kummer surface of points P, Q, and P - Q on J, the function returns the image of P + Q.
Let P and Q be points on the Jacobian J of a genus 2 curve. Given the images P1, P2, and P3 on the Kummer surface of points P, Q, P - Q on J, the function returns the image of P + n * Q.
Given the Kummer surface of the Jacobian of a genus 2 hyperelliptic curve defined over a ring R and sequence Q of three elements of R, the function returns an indexed set containing those points on K whose first three coordinates correspond to the three terms of Q.
We search for some points on the Kummer surface of the hyperelliptic curve y2=x5 - 7 defined over the rational field.
> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^5-7);
> Genus(C);
2
> J := Jacobian(C);
> K := KummerSurface(J);
> K;
Kummer surface of Jacobian of Hyperelliptic Curve defined by
y^2 = x^5 - 7 over Rational Field
> Points(K, [0,1,2]);
{@ (0 : 1 : 2 : 4) @}
> Points(K, [1,3,2]);
{@ @}
> Points(K, [0,1,3]);
{@ (0 : 1 : 3 : 9) @}
Given a point P on the Kummer surface associated to the Jacobian J (of a genus 2 curve), the function returns the indexed set of points on J mapping to P.