def local_coordinates_at_infinity(self, 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 coordinates
- ``name`` -- 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)
"""
g = self.genus()
pol = self.hyperelliptic_polynomials()[0]
K = LaurentSeriesRing(self.base_ring(), name, default_prec=prec + 2)
t = K.gen()
L = PolynomialRing(K, 'x')
x = L.gen()
i = 0
w = (x**g / t)**2 - pol
wprime = w.derivative(x)
if pol.degree() == 2 * g + 1:
x = t**-2
else:
x = t**-1
for i in range((RR(log(prec + 2) / log(2))).ceil()):
x = x - w(x) / wprime(x)
y = x**g / t
return x + O(t**(prec + 2)), y + O(t**(prec + 2))
def local_coordinates_at_infinity(self, 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 coordinates
- ``name`` -- 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)
"""
g = self.genus()
pol = self.hyperelliptic_polynomials()[0]
K = LaurentSeriesRing(self.base_ring(), name, default_prec=prec+2)
t = K.gen()
L = PolynomialRing(K,'x')
x = L.gen()
i = 0
w = (x**g/t)**2-pol
wprime = w.derivative(x)
x = t**-2
for i in range((RR(log(prec+2)/log(2))).ceil()):
x = x - w(x)/wprime(x)
y = x**g/t
return x+O(t**(prec+2)) , y+O(t**(prec+2))
def rational_points_iterator(self):
r"""
Return a generator object for the rational points on this curve.
INPUT:
- ``self`` -- a projective curve
OUTPUT:
A generator of all the rational points on the curve defined over its base field.
EXAMPLE::
sage: F = GF(37)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^7+Y*X*Z^5*55+Y^7*12)
sage: len(list(C.rational_points_iterator()))
37
::
sage: F = GF(2)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X*Y*Z)
sage: a = C.rational_points_iterator()
sage: next(a)
(1 : 0 : 0)
sage: next(a)
(0 : 1 : 0)
sage: next(a)
(1 : 1 : 0)
sage: next(a)
(0 : 0 : 1)
sage: next(a)
(1 : 0 : 1)
sage: next(a)
(0 : 1 : 1)
sage: next(a)
Traceback (most recent call last):
...
StopIteration
::
sage: F = GF(3^2,'a')
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^3+5*Y^2*Z-33*X*Y*X)
sage: b = C.rational_points_iterator()
sage: next(b)
(0 : 1 : 0)
sage: next(b)
(0 : 0 : 1)
sage: next(b)
(2*a + 2 : a : 1)
sage: next(b)
(2 : a + 1 : 1)
sage: next(b)
(a + 1 : 2*a + 1 : 1)
sage: next(b)
(1 : 2 : 1)
sage: next(b)
(2*a + 2 : 2*a : 1)
sage: next(b)
(2 : 2*a + 2 : 1)
sage: next(b)
(a + 1 : a + 2 : 1)
sage: next(b)
(1 : 1 : 1)
sage: next(b)
Traceback (most recent call last):
...
StopIteration
"""
g = self.defining_polynomial()
K = g.parent().base_ring()
from sage.rings.polynomial.all import PolynomialRing
R = PolynomialRing(K,'X')
X = R.gen()
one = K.one()
zero = K.zero()
# the point with Z = 0 = Y
try:
t = self.point([one,zero,zero])
yield(t)
except TypeError:
pass
# points with Z = 0, Y = 1
g10 = R(g(X,one,zero))
if g10.is_zero():
for x in K:
yield(self.point([x,one,zero]))
else:
for x in g10.roots(multiplicities=False):
yield(self.point([x,one,zero]))
# points with Z = 1
for y in K:
gy1 = R(g(X,y,one))
if gy1.is_zero():
for x in K:
yield(self.point([x,y,one]))
else:
for x in gy1.roots(multiplicities=False):
yield(self.point([x,y,one]))
def rational_points_iterator(self):
r"""
Return a generator object for the rational points on this curve.
INPUT:
- ``self`` -- a projective curve
OUTPUT:
A generator of all the rational points on the curve defined over its base field.
EXAMPLE::
sage: F = GF(37)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^7+Y*X*Z^5*55+Y^7*12)
sage: len(list(C.rational_points_iterator()))
37
::
sage: F = GF(2)
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X*Y*Z)
sage: a = C.rational_points_iterator()
sage: a.next()
(1 : 0 : 0)
sage: a.next()
(0 : 1 : 0)
sage: a.next()
(1 : 1 : 0)
sage: a.next()
(0 : 0 : 1)
sage: a.next()
(1 : 0 : 1)
sage: a.next()
(0 : 1 : 1)
sage: a.next()
Traceback (most recent call last):
...
StopIteration
::
sage: F = GF(3^2,'a')
sage: P2.<X,Y,Z> = ProjectiveSpace(F,2)
sage: C = Curve(X^3+5*Y^2*Z-33*X*Y*X)
sage: b = C.rational_points_iterator()
sage: b.next()
(0 : 1 : 0)
sage: b.next()
(0 : 0 : 1)
sage: b.next()
(2*a + 2 : a : 1)
sage: b.next()
(2 : a + 1 : 1)
sage: b.next()
(a + 1 : 2*a + 1 : 1)
sage: b.next()
(1 : 2 : 1)
sage: b.next()
(2*a + 2 : 2*a : 1)
sage: b.next()
(2 : 2*a + 2 : 1)
sage: b.next()
(a + 1 : a + 2 : 1)
sage: b.next()
(1 : 1 : 1)
sage: b.next()
Traceback (most recent call last):
...
StopIteration
"""
g = self.defining_polynomial()
K = g.parent().base_ring()
from sage.rings.polynomial.all import PolynomialRing
R = PolynomialRing(K, 'X')
X = R.gen()
one = K.one_element()
zero = K.zero_element()
# the point with Z = 0 = Y
try:
t = self.point([one, zero, zero])
yield (t)
except TypeError:
pass
# points with Z = 0, Y = 1
g10 = R(g(X, one, zero))
if g10.is_zero():
for x in K:
yield (self.point([x, one, zero]))
else:
for x in g10.roots(multiplicities=False):
yield (self.point([x, one, zero]))
# points with Z = 1
for y in K:
gy1 = R(g(X, y, one))
if gy1.is_zero():
for x in K:
yield (self.point([x, y, one]))
else:
for x in gy1.roots(multiplicities=False):
yield (self.point([x, y, one]))
