EXAMPLE:
sage: P.<x> = GF(5)[]
sage: f = x^5 - 3*x^4 - 2*x^3 + 6*x^2 + 3*x - 1
sage: C = HyperellipticCurve(f); C
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
EXAMPLE:
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 x0, x1 over Rational Field
Bases: sage.schemes.plane_curves.projective_curve.ProjectiveCurve_generic
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)
sage: L.<a> = K.extension(x^30-3)
sage: HK = H.change_ring(K)
sage: HL = HK.change_ring(L); HL
Hyperelliptic Curve over Eisenstein Extension of 3-adic Field with capped relative precision 5 in a defined by (1 + O(3^5))*x^30 + (O(3^6))*x^29 + (O(3^6))*x^28 + (O(3^6))*x^27 + (O(3^6))*x^26 + (O(3^6))*x^25 + (O(3^6))*x^24 + (O(3^6))*x^23 + (O(3^6))*x^22 + (O(3^6))*x^21 + (O(3^6))*x^20 + (O(3^6))*x^19 + (O(3^6))*x^18 + (O(3^6))*x^17 + (O(3^6))*x^16 + (O(3^6))*x^15 + (O(3^6))*x^14 + (O(3^6))*x^13 + (O(3^6))*x^12 + (O(3^6))*x^11 + (O(3^6))*x^10 + (O(3^6))*x^9 + (O(3^6))*x^8 + (O(3^6))*x^7 + (O(3^6))*x^6 + (O(3^6))*x^5 + (O(3^6))*x^4 + (O(3^6))*x^3 + (O(3^6))*x^2 + (O(3^6))*x + (2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + O(3^6)) 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)[]
sage: H = HyperellipticCurve(x^8 + x + 5)
sage: H.base_extend(FiniteField(7^2, 'a'))
Hyperelliptic Curve over Finite Field in a of size 7^2 defined by y^2 = x^8 + x + 5
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)
sage: L.<a> = K.extension(x^30-3)
sage: HK = H.change_ring(K)
sage: HL = HK.change_ring(L); HL
Hyperelliptic Curve over Eisenstein Extension of 3-adic Field with capped relative precision 5 in a defined by (1 + O(3^5))*x^30 + (O(3^6))*x^29 + (O(3^6))*x^28 + (O(3^6))*x^27 + (O(3^6))*x^26 + (O(3^6))*x^25 + (O(3^6))*x^24 + (O(3^6))*x^23 + (O(3^6))*x^22 + (O(3^6))*x^21 + (O(3^6))*x^20 + (O(3^6))*x^19 + (O(3^6))*x^18 + (O(3^6))*x^17 + (O(3^6))*x^16 + (O(3^6))*x^15 + (O(3^6))*x^14 + (O(3^6))*x^13 + (O(3^6))*x^12 + (O(3^6))*x^11 + (O(3^6))*x^10 + (O(3^6))*x^9 + (O(3^6))*x^8 + (O(3^6))*x^7 + (O(3^6))*x^6 + (O(3^6))*x^5 + (O(3^6))*x^4 + (O(3^6))*x^3 + (O(3^6))*x^2 + (O(3^6))*x + (2*3 + 2*3^2 + 2*3^3 + 2*3^4 + 2*3^5 + O(3^6)) 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)[]
sage: H = HyperellipticCurve(x^8 + x + 5)
sage: H.base_extend(FiniteField(7^2, 'a'))
Hyperelliptic Curve over Finite Field in a of size 7^2 defined by y^2 = x^8 + x + 5
x.__init__(...) initializes x; see help(type(x)) for signature
Return True if an odd degree model of self exists over the field of definition; False otherwise.
Use odd_degree_model to calculate an odd degree model.
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)
Returns , 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
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: set_verbose(None)
sage: H.is_singular()
False
sage: from sage.schemes.plane_curves.projective_curve import ProjectiveCurve_generic
sage: ProjectiveCurve_generic.is_singular(H)
True
Returns True, because hyperelliptic curves are smooth projective curves, as checked on construction.
EXAMPLES:
sage: R.<x> = GF(13)[]
sage: H = HyperellipticCurve(x^8+1)
sage: H.is_smooth()
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')[]
sage: H = HyperellipticCurve(x^10+2)
sage: set_verbose(None)
sage: H.is_smooth()
True
sage: from sage.schemes.plane_curves.projective_curve import ProjectiveCurve_generic
sage: ProjectiveCurve_generic.is_smooth(H)
False
x.__init__(...) initializes x; see help(type(x)) for signature
x.__init__(...) initializes x; see help(type(x)) for signature
Calls the appropriate local_coordinates function
OUTPUT: (x(t),y(t)) such that y(t)^2 = f(x(t)), where t is the local parameter at P
For the genus g hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x^g/y is the local parameter at infinity
OUTPUT: (x(t),y(t)) such that y(t)^2 = f(x(t)) and t = x^g/y is the local parameter at infinity
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)
For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter.
INPUT:
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)
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.
OUTPUT:
(x(t),y(t)) such that y(t)^2 = f(x(t)) and t = y is the local parameter at P
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)
x.__init__(...) initializes x; see help(type(x)) for signature
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')
sage: Hp2 = H.change_ring(K2).odd_degree_model(); Hp2
Hyperelliptic Curve over Number Field in a with defining polynomial x^2 + 2 defined by y^2 = 6*a*x^5 - 29*x^4 - 20*x^2 + 6*a*x + 1
sage: K3 = QuadraticField(-3, 'b')
sage: Hp3 = H.change_ring(QuadraticField(-3, 'b')).odd_degree_model(); Hp3
Hyperelliptic Curve over Number Field in b with defining polynomial x^2 + 3 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]
sage: P3 = K3.factor(43)[0][0]
sage: Hp2.change_ring(K2.residue_field(P2)).frobenius_polynomial()
x^4 - 16*x^3 + 134*x^2 - 688*x + 1849
sage: Hp3.change_ring(K3.residue_field(P3)).frobenius_polynomial()
x^4 - 16*x^3 + 134*x^2 - 688*x + 1849
sage: H.change_ring(GF(43)).odd_degree_model().frobenius_polynomial()
x^4 - 16*x^3 + 134*x^2 - 688*x + 1849
sage: P2 = K2.factor(67)[0][0]
sage: P3 = K3.factor(67)[0][0]
sage: Hp2.change_ring(K2.residue_field(P2)).frobenius_polynomial()
x^4 - 8*x^3 + 150*x^2 - 536*x + 4489
sage: Hp3.change_ring(K3.residue_field(P3)).frobenius_polynomial()
x^4 - 8*x^3 + 150*x^2 - 536*x + 4489
sage: H.change_ring(GF(67)).odd_degree_model().frobenius_polynomial()
x^4 - 8*x^3 + 150*x^2 - 536*x + 4489
sage: HyperellipticCurve(x^5 + 1, 1).odd_degree_model() Traceback (most recent call last): ... NotImplementedError: odd_degree_model only implemented for curves in Weierstrass form
sage: HyperellipticCurve(x^5 + 1, names=”U, V”).odd_degree_model() Hyperelliptic Curve over Rational Field defined by V^2 = U^5 + 1
EXAMPLES:
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: sage.schemes.hyperelliptic_curves.hyperelliptic_generic.is_HyperellipticCurve(C)
True