Hyperelliptic curves over a general ring#
EXAMPLES:
sage: P.<x> = GF(5)[] # optional - sage.rings.finite_rings
sage: f = x^5 - 3*x^4 - 2*x^3 + 6*x^2 + 3*x - 1 # optional - sage.rings.finite_rings
sage: C = HyperellipticCurve(f); C # optional - sage.rings.finite_rings
Hyperelliptic Curve over Finite Field of size 5
defined by y^2 = x^5 + 2*x^4 + 3*x^3 + x^2 + 3*x + 4
sage: P.<x> = QQ[]
sage: f = 4*x^5 - 30*x^3 + 45*x - 22
sage: C = HyperellipticCurve(f); C
Hyperelliptic Curve over Rational Field defined by y^2 = 4*x^5 - 30*x^3 + 45*x - 22
sage: C.genus()
2
sage: D = C.affine_patch(0)
sage: D.defining_polynomials()[0].parent()
Multivariate Polynomial Ring in x1, x2 over Rational Field
- class sage.schemes.hyperelliptic_curves.hyperelliptic_generic.HyperellipticCurve_generic(PP, f, h=None, names=None, genus=None)#
Bases:
ProjectivePlaneCurve- base_extend(R)#
Returns this HyperellipticCurve over a new base ring
R.EXAMPLES:
sage: R.<x> = QQ[] sage: H = HyperellipticCurve(x^5 - 10*x + 9) sage: K = Qp(3, 5) # optional - sage.rings.padics sage: L.<a> = K.extension(x^30 - 3) # optional - sage.rings.padics sage: HK = H.change_ring(K) # optional - sage.rings.padics sage: HL = HK.change_ring(L); HL # optional - sage.rings.padics Hyperelliptic Curve over 3-adic Eisenstein Extension Field in a defined by x^30 - 3 defined by (1 + O(a^150))*y^2 = (1 + O(a^150))*x^5 + (2 + 2*a^30 + a^60 + 2*a^90 + 2*a^120 + O(a^150))*x + a^60 + O(a^210) sage: R.<x> = FiniteField(7)[] # optional - sage.rings.finite_rings sage: H = HyperellipticCurve(x^8 + x + 5) # optional - sage.rings.finite_rings sage: H.base_extend(FiniteField(7^2, 'a')) # optional - sage.rings.finite_rings Hyperelliptic Curve over Finite Field in a of size 7^2 defined by y^2 = x^8 + x + 5
- change_ring(R)#
Returns this HyperellipticCurve over a new base ring
R.EXAMPLES:
sage: R.<x> = QQ[] sage: H = HyperellipticCurve(x^5 - 10*x + 9) sage: K = Qp(3, 5) # optional - sage.rings.padics sage: L.<a> = K.extension(x^30 - 3) # optional - sage.rings.padics sage: HK = H.change_ring(K) # optional - sage.rings.padics sage: HL = HK.change_ring(L); HL # optional - sage.rings.padics Hyperelliptic Curve over 3-adic Eisenstein Extension Field in a defined by x^30 - 3 defined by (1 + O(a^150))*y^2 = (1 + O(a^150))*x^5 + (2 + 2*a^30 + a^60 + 2*a^90 + 2*a^120 + O(a^150))*x + a^60 + O(a^210) sage: R.<x> = FiniteField(7)[] # optional - sage.rings.finite_rings sage: H = HyperellipticCurve(x^8 + x + 5) # optional - sage.rings.finite_rings sage: H.base_extend(FiniteField(7^2, 'a')) # optional - sage.rings.finite_rings Hyperelliptic Curve over Finite Field in a of size 7^2 defined by y^2 = x^8 + x + 5
- genus()#
- has_odd_degree_model()#
Return True if an odd degree model of self exists over the field of definition; False otherwise.
Use
odd_degree_modelto calculate an odd degree model.EXAMPLES:
sage: x = QQ['x'].0 sage: HyperellipticCurve(x^5 + x).has_odd_degree_model() True sage: HyperellipticCurve(x^6 + x).has_odd_degree_model() True sage: HyperellipticCurve(x^6 + x + 1).has_odd_degree_model() False
- hyperelliptic_polynomials(K=None, var='x')#
EXAMPLES:
sage: R.<x> = QQ[]; C = HyperellipticCurve(x^3 + x - 1, x^3/5); C Hyperelliptic Curve over Rational Field defined by y^2 + 1/5*x^3*y = x^3 + x - 1 sage: C.hyperelliptic_polynomials() (x^3 + x - 1, 1/5*x^3)
- invariant_differential()#
Returns \(dx/2y\), as an element of the Monsky-Washnitzer cohomology of self
EXAMPLES:
sage: R.<x> = QQ['x'] sage: C = HyperellipticCurve(x^5 - 4*x + 4) sage: C.invariant_differential() 1 dx/2y
- is_singular()#
Returns False, because hyperelliptic curves are smooth projective curves, as checked on construction.
EXAMPLES:
sage: R.<x> = QQ[] sage: H = HyperellipticCurve(x^5 + 1) sage: H.is_singular() False
A hyperelliptic curve with genus at least 2 always has a singularity at infinity when viewed as a plane projective curve. This can be seen in the following example.:
sage: R.<x> = QQ[] sage: H = HyperellipticCurve(x^5 + 2) sage: from sage.misc.verbose import set_verbose sage: set_verbose(-1) sage: H.is_singular() False sage: from sage.schemes.curves.projective_curve import ProjectivePlaneCurve sage: ProjectivePlaneCurve.is_singular(H) True
- is_smooth()#
Returns True, because hyperelliptic curves are smooth projective curves, as checked on construction.
EXAMPLES:
sage: R.<x> = GF(13)[] # optional - sage.rings.finite_rings sage: H = HyperellipticCurve(x^8 + 1) # optional - sage.rings.finite_rings sage: H.is_smooth() # optional - sage.rings.finite_rings True
A hyperelliptic curve with genus at least 2 always has a singularity at infinity when viewed as a plane projective curve. This can be seen in the following example.:
sage: R.<x> = GF(27, 'a')[] # optional - sage.rings.finite_rings sage: H = HyperellipticCurve(x^10 + 2) # optional - sage.rings.finite_rings sage: from sage.misc.verbose import set_verbose sage: set_verbose(-1) sage: H.is_smooth() # optional - sage.rings.finite_rings True sage: from sage.schemes.curves.projective_curve import ProjectivePlaneCurve sage: ProjectivePlaneCurve.is_smooth(H) # optional - sage.rings.finite_rings False
- jacobian()#
- lift_x(x, all=False)#
- local_coord(P, prec=20, name='t')#
Calls the appropriate local_coordinates function
INPUT:
P– a point on selfprec– desired precision of the local coordinatesname– generator of the power series ring (default:t)
OUTPUT:
\((x(t),y(t))\) such that \(y(t)^2 = f(x(t))\), where \(t\) is the local parameter at \(P\)
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^5 - 23*x^3 + 18*x^2 + 40*x) sage: H.local_coord(H(1 ,6), prec=5) (1 + t + O(t^5), 6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5)) sage: H.local_coord(H(4, 0), prec=7) (4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7), t + O(t^7)) sage: H.local_coord(H(0, 1, 0), prec=5) (t^-2 + 23*t^2 - 18*t^4 - 569*t^6 + O(t^7), t^-5 + 46*t^-1 - 36*t - 609*t^3 + 1656*t^5 + O(t^6))
AUTHOR:
Jennifer Balakrishnan (2007-12)
- local_coordinates_at_infinity(prec=20, name='t')#
For the genus \(g\) hyperelliptic curve \(y^2 = f(x)\), return \((x(t), y(t))\) such that \((y(t))^2 = f(x(t))\), where \(t = x^g/y\) is the local parameter at infinity
INPUT:
prec– desired precision of the local coordinatesname– generator of the power series ring (default:t)
OUTPUT:
\((x(t),y(t))\) such that \(y(t)^2 = f(x(t))\) and \(t = x^g/y\) is the local parameter at infinity
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^5 - 5*x^2 + 1) sage: x, y = H.local_coordinates_at_infinity(10) sage: x t^-2 + 5*t^4 - t^8 - 50*t^10 + O(t^12) sage: y t^-5 + 10*t - 2*t^5 - 75*t^7 + 50*t^11 + O(t^12)
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^3 - x + 1) sage: x, y = H.local_coordinates_at_infinity(10) sage: x t^-2 + t^2 - t^4 - t^6 + 3*t^8 + O(t^12) sage: y t^-3 + t - t^3 - t^5 + 3*t^7 - 10*t^11 + O(t^12)
AUTHOR:
Jennifer Balakrishnan (2007-12)
- local_coordinates_at_nonweierstrass(P, prec=20, name='t')#
For a non-Weierstrass point \(P = (a,b)\) on the hyperelliptic curve \(y^2 = f(x)\), return \((x(t), y(t))\) such that \((y(t))^2 = f(x(t))\), where \(t = x - a\) is the local parameter.
INPUT:
P = (a, b)– a non-Weierstrass point on selfprec– desired precision of the local coordinatesname– gen of the power series ring (default:t)
OUTPUT:
\((x(t),y(t))\) such that \(y(t)^2 = f(x(t))\) and \(t = x - a\) is the local parameter at \(P\)
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^5 - 23*x^3 + 18*x^2 + 40*x) sage: P = H(1, 6) sage: x, y = H.local_coordinates_at_nonweierstrass(P, prec=5) sage: x 1 + t + O(t^5) sage: y 6 + t - 7/2*t^2 - 1/2*t^3 - 25/48*t^4 + O(t^5) sage: Q = H(-2, 12) sage: x, y = H.local_coordinates_at_nonweierstrass(Q, prec=5) sage: x -2 + t + O(t^5) sage: y 12 - 19/2*t - 19/32*t^2 + 61/256*t^3 - 5965/24576*t^4 + O(t^5)
AUTHOR:
Jennifer Balakrishnan (2007-12)
- local_coordinates_at_weierstrass(P, prec=20, name='t')#
For a finite Weierstrass point on the hyperelliptic curve \(y^2 = f(x)\), returns \((x(t), y(t))\) such that \((y(t))^2 = f(x(t))\), where \(t = y\) is the local parameter.
INPUT:
P– a finite Weierstrass point on selfprec– desired precision of the local coordinatesname– gen of the power series ring (default: \(t\))
OUTPUT:
\((x(t),y(t))\) such that \(y(t)^2 = f(x(t))\) and \(t = y\) is the local parameter at \(P\)
EXAMPLES:
sage: R.<x> = QQ['x'] sage: H = HyperellipticCurve(x^5 - 23*x^3 + 18*x^2 + 40*x) sage: A = H(4, 0) sage: x, y = H.local_coordinates_at_weierstrass(A, prec=7) sage: x 4 + 1/360*t^2 - 191/23328000*t^4 + 7579/188956800000*t^6 + O(t^7) sage: y t + O(t^7) sage: B = H(-5, 0) sage: x, y = H.local_coordinates_at_weierstrass(B, prec=5) sage: x -5 + 1/1260*t^2 + 887/2000376000*t^4 + O(t^5) sage: y t + O(t^5)
- AUTHOR:
Jennifer Balakrishnan (2007-12)
Francis Clarke (2012-08-26)
- monsky_washnitzer_gens()#
- odd_degree_model()#
Return an odd degree model of self, or raise ValueError if one does not exist over the field of definition.
EXAMPLES:
sage: x = QQ['x'].gen() sage: H = HyperellipticCurve((x^2 + 2)*(x^2 + 3)*(x^2 + 5)); H Hyperelliptic Curve over Rational Field defined by y^2 = x^6 + 10*x^4 + 31*x^2 + 30 sage: H.odd_degree_model() Traceback (most recent call last): ... ValueError: No odd degree model exists over field of definition sage: K2 = QuadraticField(-2, 'a') # optional - sage.rings.number_field sage: Hp2 = H.change_ring(K2).odd_degree_model(); Hp2 # optional - sage.rings.number_field Hyperelliptic Curve over Number Field in a with defining polynomial x^2 + 2 with a = 1.414213562373095?*I defined by y^2 = 6*a*x^5 - 29*x^4 - 20*x^2 + 6*a*x + 1 sage: K3 = QuadraticField(-3, 'b') # optional - sage.rings.number_field sage: Hp3 = H.change_ring(QuadraticField(-3, 'b')).odd_degree_model(); Hp3 # optional - sage.rings.number_field Hyperelliptic Curve over Number Field in b with defining polynomial x^2 + 3 with b = 1.732050807568878?*I defined by y^2 = -4*b*x^5 - 14*x^4 - 20*b*x^3 - 35*x^2 + 6*b*x + 1 Of course, ``Hp2`` and ``Hp3`` are isomorphic over the composite extension. One consequence of this is that odd degree models reduced over "different" fields should have the same number of points on their reductions. 43 and 67 split completely in the compositum, so when we reduce we find: sage: P2 = K2.factor(43)[0][0] # optional - sage.rings.number_field sage: P3 = K3.factor(43)[0][0] # optional - sage.rings.number_field sage: Hp2.change_ring(K2.residue_field(P2)).frobenius_polynomial() # optional - sage.rings.number_field x^4 - 16*x^3 + 134*x^2 - 688*x + 1849 sage: Hp3.change_ring(K3.residue_field(P3)).frobenius_polynomial() # optional - sage.rings.number_field x^4 - 16*x^3 + 134*x^2 - 688*x + 1849 sage: H.change_ring(GF(43)).odd_degree_model().frobenius_polynomial() # optional - sage.rings.finite_rings x^4 - 16*x^3 + 134*x^2 - 688*x + 1849 sage: P2 = K2.factor(67)[0][0] # optional - sage.rings.number_field sage: P3 = K3.factor(67)[0][0] # optional - sage.rings.number_field sage: Hp2.change_ring(K2.residue_field(P2)).frobenius_polynomial() # optional - sage.rings.number_field x^4 - 8*x^3 + 150*x^2 - 536*x + 4489 sage: Hp3.change_ring(K3.residue_field(P3)).frobenius_polynomial() # optional - sage.rings.number_field x^4 - 8*x^3 + 150*x^2 - 536*x + 4489 sage: H.change_ring(GF(67)).odd_degree_model().frobenius_polynomial() # optional - sage.rings.finite_rings x^4 - 8*x^3 + 150*x^2 - 536*x + 4489
- rational_points(**kwds)#
Find rational points on the hyperelliptic curve, all arguments are passed on to
sage.schemes.generic.algebraic_scheme.rational_points().EXAMPLES:
For the LMFDB genus 2 curve 932.a.3728.1:
sage: R.<x> = PolynomialRing(QQ) sage: C = HyperellipticCurve(R([0, -1, 1, 0, 1, -2, 1]), R([1])) sage: C.rational_points(bound=8) [(-1 : -3 : 1), (-1 : 2 : 1), (0 : -1 : 1), (0 : 0 : 1), (0 : 1 : 0), (1/2 : -5/8 : 1), (1/2 : -3/8 : 1), (1 : -1 : 1), (1 : 0 : 1)]
Check that github issue #29509 is fixed for the LMFDB genus 2 curve 169.a.169.1:
sage: C = HyperellipticCurve(R([0, 0, 0, 0, 1, 1]), R([1, 1, 0, 1])) sage: C.rational_points(bound=10) [(-1 : 0 : 1), (-1 : 1 : 1), (0 : -1 : 1), (0 : 0 : 1), (0 : 1 : 0)]
An example over a number field:
sage: R.<x> = PolynomialRing(QuadraticField(2)) # optional - sage.rings.number_field sage: C = HyperellipticCurve(R([1, 0, 0, 0, 0, 1])) # optional - sage.rings.number_field sage: C.rational_points(bound=2) # optional - sage.rings.number_field [(-1 : 0 : 1), (0 : -1 : 1), (0 : 1 : 0), (0 : 1 : 1), (1 : -a : 1), (1 : a : 1)]
- sage.schemes.hyperelliptic_curves.hyperelliptic_generic.is_HyperellipticCurve(C)#
EXAMPLES:
sage: from sage.schemes.hyperelliptic_curves.hyperelliptic_generic import is_HyperellipticCurve sage: R.<x> = QQ[]; C = HyperellipticCurve(x^3 + x - 1); C Hyperelliptic Curve over Rational Field defined by y^2 = x^3 + x - 1 sage: is_HyperellipticCurve(C) True