# Rational point sets on a Jacobian¶

EXAMPLES:

sage: x = QQ['x'].0
sage: f = x^5 + x + 1
sage: C = HyperellipticCurve(f); C
Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: C(QQ)
Set of rational points of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: P = C([0,1,1])
sage: J = C.jacobian(); J
Jacobian of Hyperelliptic Curve over Rational Field defined by y^2 = x^5 + x + 1
sage: Q = J(QQ)(P); Q
(x, y - 1)
sage: Q + Q
(x^2, y - 1/2*x - 1)
sage: Q*3
(x^2 - 1/64*x + 1/8, y + 255/512*x + 65/64)

sage: F.<a> = GF(3)
sage: R.<x> = F[]
sage: f = x^5-1
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: X = J(F)
sage: a = x^2-x+1
sage: b = -x +1
sage: c = x-1
sage: d = 0
sage: D1 = X([a,b])
sage: D1
(x^2 + 2*x + 1, y + x + 2)
sage: D2 = X([c,d])
sage: D2
(x + 2, y)
sage: D1+D2
(x^2 + 2*x + 2, y + 2*x + 1)

class sage.schemes.hyperelliptic_curves.jacobian_homset.JacobianHomset_divisor_classes(Y, X, **kwds)
base_extend(R)

x.__init__(...) initializes x; see help(type(x)) for signature

curve()

x.__init__(...) initializes x; see help(type(x)) for signature

value_ring()

Returns S for a homset X(T) where T = Spec(S).

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Jacobian of a General Hyperelliptic Curve

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Jacobian ‘morphism’ as a class in the Picard group