# Jacobian of a General Hyperelliptic Curve¶

class sage.schemes.hyperelliptic_curves.jacobian_generic.HyperellipticJacobian_generic(C)

Bases: sage.schemes.jacobians.abstract_jacobian.Jacobian_generic

EXAMPLES:

sage: FF = FiniteField(2003)
sage: R.<x> = PolynomialRing(FF)
sage: f = x**5 + 1184*x**3 + 1846*x**2 + 956*x + 560
sage: C = HyperellipticCurve(f)
sage: J = C.jacobian()
sage: a = x**2 + 376*x + 245; b = 1015*x + 1368
sage: X = J(FF)
sage: D = X([a,b])
sage: D
(x^2 + 376*x + 245, y + 988*x + 635)
sage: J(0)
(1)
sage: D == J([a,b])
True
sage: D == D + J(0)
True

An more extended example, demonstrating arithmetic in J(QQ) and J(K) for a number field K/QQ.

sage: P.<x> = PolynomialRing(QQ)
sage: f = x^5 - x + 1; h = x
sage: C = HyperellipticCurve(f,h,'u,v')
sage: C
Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
sage: PP = C.ambient_space()
sage: PP
Projective Space of dimension 2 over Rational Field
sage: C.defining_polynomial()
-x0^5 + x0*x1*x2^3 + x1^2*x2^3 + x0*x2^4 - x2^5
sage: C(QQ)
Set of rational points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
sage: K.<t> = NumberField(x^2-2)
sage: C(K)
Set of rational points of Closed subscheme of Projective Space of dimension 2 over Number Field in t with defining polynomial x^2 - 2 defined by: -x0^5 + x0*x1*x2^3 + x1^2*x2^3 + x0*x2^4 - x2^5
sage: P = C(QQ)(0,1,1); P
(0 : 1 : 1)
sage: P == C(0,1,1)
True
sage: C(0,1,1).parent()
Set of rational points of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
sage: P1 = C(K)(P)
sage: P2 = C(K)([2,4*t-1,1])
sage: P3 = C(K)([-1/2,1/8*(7*t+2),1])
sage: P1, P2, P3
((0 : 1 : 1), (2 : 4*t - 1 : 1), (-1/2 : 7/8*t + 1/4 : 1))
sage: J = C.jacobian()
sage: J
Jacobian of Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1
sage: Q = J(QQ)(P); Q
(u, v - 1)
sage: for i in range(6): Q*i
(1)
(u, v - 1)
(u^2, v + u - 1)
(u^2, v + 1)
(u, v + 1)
(1)
sage: Q1 = J(K)(P1); print "%s -> %s"%( P1, Q1 )
(0 : 1 : 1) -> (u, v - 1)
sage: Q2 = J(K)(P2); print "%s -> %s"%( P2, Q2 )
(2 : 4*t - 1 : 1) -> (u - 2, v - 4*t + 1)
sage: Q3 = J(K)(P3); print "%s -> %s"%( P3, Q3 )
(-1/2 : 7/8*t + 1/4 : 1) -> (u + 1/2, v - 7/8*t - 1/4)
sage: R.<x> = PolynomialRing(K)
sage: Q4 = J(K)([x^2-t,R(1)])
sage: for i in range(4): Q4*i
(1)
(u^2 - t, v - 1)
(u^2 + (-3/4*t - 9/16)*u + 1/2*t + 1/4, v + (-1/32*t - 57/64)*u + 1/2*t + 9/16)
(u^2 + (1352416/247009*t - 1636930/247009)*u - 1156544/247009*t + 1900544/247009, v + (-2326345442/122763473*t + 3233153137/122763473)*u + 2439343104/122763473*t - 3350862929/122763473)
sage: R2 = Q2*5; R2
(u^2 - 3789465233/116983808*u - 267915823/58491904, v + (-233827256513849/1789384327168*t + 1/2)*u - 15782925357447/894692163584*t)
sage: R3 = Q3*5; R3
(u^2 + 5663300808399913890623/14426454798950909645952*u - 26531814176395676231273/28852909597901819291904, v + (253155440321645614070860868199103/2450498420175733688903836378159104*t + 1/2)*u + 2427708505064902611513563431764311/4900996840351467377807672756318208*t)
sage: R4 = Q4*5; R4
(u^2 - 3789465233/116983808*u - 267915823/58491904, v + (233827256513849/1789384327168*t + 1/2)*u + 15782925357447/894692163584*t)
sage: # Thus we find the following identity:
sage: 5*Q2 + 5*Q4
(1)
sage: # Moreover the following relation holds in the 5-torsion subgroup:
sage: Q2 + Q4 == 2*Q1
True
dimension()

Return the dimension of this Jacobian.

OUTPUT:
Integer

EXAMPLES:

sage: k.<a> = GF(9); R.<x> = k[]
sage: HyperellipticCurve(x^3 + x - 1, x+a).jacobian().dimension()
1
sage: g = HyperellipticCurve(x^6 + x - 1, x+a).jacobian().dimension(); g
2
sage: type(g)
<type 'sage.rings.integer.Integer'>
point(mumford, check=True)

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

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Jacobian of a Hyperelliptic curve of Genus 2

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Rational point sets on a Jacobian