The hyperelliptic curve is embedded in a weighted projective space, with weights 1, g + 1, and 1, respectively on x, y and z. Therefore point triples satisfy the equivalence relation (x : y : z) = (mu x : mu^(g + 1) y : mu z), and the points at infinity are then normalized to take the form (1 : y : 0).
Returns the point on a hyperelliptic curve C specified by the coordinates (x, y, z). The elt constructor takes the pointset of a hyperelliptic curve as an argument. If z is not specified it is assumed to be 1.
Given a point P on a hyperelliptic curve C_1, such that C is a base extension of C_1, returns the corresponding point on the hyperelliptic curve C. The curve C can be, e.g., the reduction of C_1 to finite characteristic (i.e. base extension to a finite field) or the tautological coercion to itself.
The indexed set of all rational points on the hyperelliptic curve C that have the value x as their x-coordinate.
The points at infinity for the hyperelliptic curve C returned as an indexed set of points.
Returns true if and only if the sequence S specifies a point on the hyperelliptic curve C, and if so, returns this point as the second value.
Given a hyperelliptic curve C defined over a finite field, returns a point chosen at random on the curve. If the set of all points on C has already been computed, this gives a truly random point, otherwise the ramification points have a slight advantage.
Returns true if and only if the two points P and Q on the same hyperelliptic curve have the same coordinates.
Returns false if and only if the two points P and Q on the same hyperelliptic curve have the same coordinates.
The i-th coordinate of the point P, for 1 <= i <= 3.
Given a point P on a hyperelliptic curve, returns a 3-element sequence consisting of the coordinates of the point P.
Given a point P on a hyperelliptic curve, returns the image of P under the hyperelliptic involution.
Given a hyperelliptic curve C defined over a finite field, returns the number of rational points on C.
Bound: RngIntElt Default:
For a hyperelliptic curve C defined over a finite field returns all rational points on C. For a curve C over Q of the form y^2 = f(x) with integral coefficients, returns the set of points such that the naive height of the x-coordinate is less than Bound.
For a hyperelliptic curve C defined over a finite field returns all rational points on C.
Returns true if and only if the points of C have been computed. This can especially be helpful when the curve is likely to have many points and when one does not wish to trigger the possibly expensive point computation.
Given a hyperelliptic curve C defined over a finite field, this function computes the zeta function of C. The zeta function is returned as an element of the function field in one variable over the integers.
Given a hyperelliptic curve C defined over the rationals, this function computes the zeta function of the base extension of C to K. The curve C must have good reduction at the characteristic of K.
> P<x> := PolynomialRing(Rationals());
> C := HyperellipticCurve(x^6+x^2+1);
> Points(C : Bound := 1);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1) @}
> Points(C : Bound := 2);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1), (-1 : -9 : 2),
(-1 : 9 : 2), (1 : -9 : 2), (1 : 9 : 2) @}
> Points(C : Bound := 4);
{@ (1 : -1 : 0), (1 : 1 : 0), (0 : -1 : 1), (0 : 1 : 1), (-1 : -9 : 2),
(-1 : 9 : 2), (1 : -9 : 2), (1 : 9 : 2) @}
Check: BoolElt Default: true
Applies the Frobenius x - > x^((#F)) to P. If Check is true, it verifies that the curve of which P is a point is defined over F.[Next][Prev] [Right] [Left] [Up] [Index] [Root]