Jacobian of a general hyperelliptic curve#
- class sage.schemes.hyperelliptic_curves.jacobian_generic.HyperellipticJacobian_generic(C)#
Bases:
Jacobian_genericEXAMPLES:
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 beFalseuntil 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
selfis 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 beFalseuntil 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)#