Jacobian of a general hyperelliptic curve#
- class sage.schemes.hyperelliptic_curves.jacobian_generic.HyperellipticJacobian_generic(C)#
Bases:
Jacobian_generic
EXAMPLES:
sage: # needs sage.rings.finite_rings 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
A 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'); C Hyperelliptic Curve over Rational Field defined by v^2 + u*v = u^5 - u + 1 sage: PP = C.ambient_space(); 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) # needs sage.rings.number_field sage: C(K) # needs sage.rings.number_field Set of rational points of Hyperelliptic Curve over Number Field in t with defining polynomial x^2 - 2 defined by v^2 + u*v = u^5 - u + 1 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: # needs sage.rings.number_field 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(); 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: # needs sage.rings.number_field 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)
Thus we find the following identity:
sage: 5*Q2 + 5*Q4 # needs sage.rings.number_field (1)
Moreover the following relation holds in the 5-torsion subgroup:
sage: Q2 + Q4 == 2*Q1 # needs sage.rings.number_field True
- dimension()#
Return the dimension of this Jacobian.
OUTPUT:
Integer
EXAMPLES:
sage: # needs sage.rings.finite_rings 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) <... 'sage.rings.integer.Integer'>
- geometric_endomorphism_algebra_is_field(B=200, proof=False)#
Return whether the geometric endomorphism algebra is a field.
This implies that the Jacobian of the curve is geometrically simple. It is based on Algorithm 4.10 from [Lom2019]
INPUT:
B
– (default: 200) the bound which appears in the statement of the algorithm from [Lom2019]proof
– (default: False) whether or not to insist on a provably correct answer. This is related to the warning in the docstring of this module: if this function returnsFalse
, then strictly speaking this has not been proven to beFalse
until one has exhibited a non-trivial endomorphism, which these methods are not designed to carry out. If one is convinced that this method should returnTrue
, but it is returningFalse
, then this can be exhibited by increasing \(B\).
OUTPUT:
Boolean indicating whether or not the geometric endomorphism algebra is a field.
EXAMPLES:
This is LMFDB curve 262144.d.524288.2 which has QM. Although its Jacobian is geometrically simple, the geometric endomorphism algebra is not a field:
sage: R.<x> = QQ[] sage: f = x^5 + x^4 + 4*x^3 + 8*x^2 + 5*x + 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_algebra_is_field() False
This is LMFDB curve 50000.a.200000.1:
sage: f = 8*x^5 + 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_algebra_is_field() True
- geometric_endomorphism_ring_is_ZZ(B=200, proof=False)#
Return whether the geometric endomorphism ring of
self
is the integer ring \(\ZZ\).INPUT:
B
– (default: 200) the bound which appears in the statement of the algorithm from [Lom2019]proof
– (default: False) whether or not to insist on a provably correct answer. This is related to the warning in the module docstring of \(jacobian_endomorphisms.py\): if this function returnsFalse
, then strictly speaking this has not been proven to beFalse
until one has exhibited a non-trivial endomorphism, which the methods in that module are not designed to carry out. If one is convinced that this method should returnTrue
, but it is returningFalse
, then this can be exhibited by increasing \(B\).
OUTPUT:
Boolean indicating whether or not the geometric endomorphism ring is isomorphic to the integer ring.
EXAMPLES:
This is LMFDB curve 603.a.603.2:
sage: R.<x> = QQ[] sage: f = 4*x^5 + x^4 - 4*x^3 + 2*x^2 + 4*x + 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() True
This is LMFDB curve 1152.a.147456.1 whose geometric endomorphism ring is isomorphic to the group of 2x2 matrices over \(\QQ\):
sage: f = x^6 - 2*x^4 + 2*x^2 - 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
This is LMFDB curve 20736.k.373248.1 whose geometric endomorphism ring is isomorphic to the group of 2x2 matrices over a CM field:
sage: f = x^6 + 8 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
This is LMFDB curve 708.a.181248.1:
sage: R.<x> = QQ[] sage: f = -3*x^6 - 16*x^5 + 36*x^4 + 194*x^3 - 164*x^2 - 392*x - 143 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() True
This is LMFDB curve 10609.a.10609.1 whose geometric endomorphism ring is an order in a real quadratic field:
sage: f = x^6 + 2*x^4 + 2*x^3 + 5*x^2 + 6*x + 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
This is LMFDB curve 160000.c.800000.1 whose geometric endomorphism ring is an order in a CM field:
sage: f = x^5 - 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
This is LMFDB curve 262144.d.524288.2 whose geometric endomorphism ring is an order in a quaternion algebra:
sage: f = x^5 + x^4 + 4*x^3 + 8*x^2 + 5*x + 1 sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
This is LMFDB curve 578.a.2312.1 whose geometric endomorphism ring is \(\QQ \times \QQ\):
sage: f = 4*x^5 - 7*x^4 + 10*x^3 - 7*x^2 + 4*x sage: C = HyperellipticCurve(f) sage: J = C.jacobian() sage: J.geometric_endomorphism_ring_is_ZZ() False
- point(mumford, check=True)#