def curve_over_ram_extn(self, deg):
r"""
Returns self over $\Q_p(p^(1/deg))$
INPUT:
- deg: the degree of the ramified extension
OUTPUT:
self over $\Q_p(p^(1/deg))$
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: K = Qp(11,5)
sage: HK = H.change_ring(K)
sage: HL = HK.curve_over_ram_extn(2)
sage: HL
Hyperelliptic Curve over Eisenstein Extension of 11-adic Field with capped relative precision 5 in a defined by (1 + O(11^5))*x^2 + (O(11^6))*x + (10*11 + 10*11^2 + 10*11^3 + 10*11^4 + 10*11^5 + O(11^6)) defined by (1 + O(a^10))*y^2 = (1 + O(a^10))*x^5 + (10 + 8*a^2 + 10*a^4 + 10*a^6 + 10*a^8 + O(a^10))*x^3 + (7 + a^2 + O(a^10))*x^2 + (7 + 3*a^2 + O(a^10))*x
AUTHOR:
- Jennifer Balakrishnan
"""
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
K = self.base_ring()
p = K.prime()
A = PolynomialRing(QQ, "x")
x = A.gen()
J = K.extension(x ** deg - p, names="a")
pol = self.hyperelliptic_polynomials()[0]
H = HyperellipticCurve(A(pol))
HJ = H.change_ring(J)
self._curve_over_ram_extn = HJ
self._curve_over_ram_extn._curve_over_Qp = self
return HJ
def curve_over_ram_extn(self, deg):
r"""
Return ``self`` over `\QQ_p(p^(1/deg))`.
INPUT:
- deg: the degree of the ramified extension
OUTPUT:
``self`` over `\QQ_p(p^(1/deg))`
EXAMPLES::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^5-23*x^3+18*x^2+40*x)
sage: K = Qp(11,5)
sage: HK = H.change_ring(K)
sage: HL = HK.curve_over_ram_extn(2)
sage: HL
Hyperelliptic Curve over Eisenstein Extension in a defined by x^2 - 11 with capped relative precision 10 over 11-adic Field defined by (1 + O(a^10))*y^2 = (1 + O(a^10))*x^5 + (10 + 8*a^2 + 10*a^4 + 10*a^6 + 10*a^8 + O(a^10))*x^3 + (7 + a^2 + O(a^10))*x^2 + (7 + 3*a^2 + O(a^10))*x
AUTHOR:
- Jennifer Balakrishnan
"""
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
K = self.base_ring()
p = K.prime()
A = PolynomialRing(QQ, 'x')
x = A.gen()
J = K.extension(x**deg - p, names='a')
pol = self.hyperelliptic_polynomials()[0]
H = HyperellipticCurve(A(pol))
HJ = H.change_ring(J)
self._curve_over_ram_extn = HJ
self._curve_over_ram_extn._curve_over_Qp = self
return HJ
def coleman_integrals_on_basis(self, P, Q, algorithm=None):
r"""
Computes the Coleman integrals `\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}`
INPUT:
- P point on self
- Q point on self
- algorithm (optional) = None (uses Frobenius) or teichmuller (uses Teichmuller points)
OUTPUT:
the Coleman integrals `\{\int_P^Q x^i dx/2y \}_{i=0}^{2g-1}`
EXAMPLES::
sage: K = pAdicField(11, 5)
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^5 + 33/16*x^4 + 3/4*x^3 + 3/8*x^2 - 1/4*x + 1/16)
sage: P = C.lift_x(2)
sage: Q = C.lift_x(3)
sage: C.coleman_integrals_on_basis(P, Q)
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
sage: C.coleman_integrals_on_basis(P, Q, algorithm='teichmuller')
(10*11 + 6*11^3 + 2*11^4 + O(11^5), 11 + 9*11^2 + 7*11^3 + 9*11^4 + O(11^5), 3 + 10*11 + 5*11^2 + 9*11^3 + 4*11^4 + O(11^5), 3 + 11 + 5*11^2 + 4*11^4 + O(11^5))
::
sage: K = pAdicField(11,5)
sage: x = polygen(K)
sage: C = HyperellipticCurve(x^5 + 33/16*x^4 + 3/4*x^3 + 3/8*x^2 - 1/4*x + 1/16)
sage: P = C.lift_x(11^(-2))
sage: Q = C.lift_x(3*11^(-2))
sage: C.coleman_integrals_on_basis(P, Q)
(3*11^3 + 7*11^4 + 4*11^5 + 7*11^6 + 5*11^7 + O(11^8), 3*11 + 10*11^2 + 8*11^3 + 9*11^4 + 7*11^5 + O(11^6), 4*11^-1 + 2 + 6*11 + 6*11^2 + 7*11^3 + O(11^4), 11^-3 + 6*11^-2 + 2*11^-1 + 2 + O(11^2))
::
sage: R = C(0,1/4)
sage: a = C.coleman_integrals_on_basis(P,R) # long time (7s on sage.math, 2011)
sage: b = C.coleman_integrals_on_basis(R,Q) # long time (9s on sage.math, 2011)
sage: c = C.coleman_integrals_on_basis(P,Q) # long time
sage: a+b == c # long time
True
::
sage: R.<x> = QQ['x']
sage: H = HyperellipticCurve(x^3-10*x+9)
sage: K = Qp(5,8)
sage: HK = H.change_ring(K)
sage: S = HK(1,0)
sage: P = HK(0,3)
sage: T = HK(0,1,0)
sage: Q = HK.lift_x(5^-2)
sage: R = HK.lift_x(4*5^-2)
sage: HK.coleman_integrals_on_basis(S,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(T,P)
(2*5^2 + 5^4 + 5^5 + 3*5^6 + 3*5^7 + 2*5^8 + O(5^9), 5 + 2*5^2 + 4*5^3 + 2*5^4 + 3*5^6 + 4*5^7 + 2*5^8 + O(5^9))
sage: HK.coleman_integrals_on_basis(P,S) == -HK.coleman_integrals_on_basis(S,P)
True
sage: HK.coleman_integrals_on_basis(S,Q)
(4*5 + 4*5^2 + 4*5^3 + O(5^4), 5^-1 + O(5^3))
sage: HK.coleman_integrals_on_basis(Q,R)
(4*5 + 2*5^2 + 2*5^3 + 2*5^4 + 5^5 + 5^6 + 5^7 + 3*5^8 + O(5^9), 2*5^-1 + 4 + 4*5 + 4*5^2 + 4*5^3 + 2*5^4 + 3*5^5 + 2*5^6 + O(5^7))
sage: HK.coleman_integrals_on_basis(S,R) == HK.coleman_integrals_on_basis(S,Q) + HK.coleman_integrals_on_basis(Q,R)
True
sage: HK.coleman_integrals_on_basis(T,T)
(0, 0)
sage: HK.coleman_integrals_on_basis(S,T)
(0, 0)
AUTHORS:
- Robert Bradshaw (2007-03): non-Weierstrass points
- Jennifer Balakrishnan and Robert Bradshaw (2010-02): Weierstrass points
"""
import sage.schemes.hyperelliptic_curves.monsky_washnitzer as monsky_washnitzer
from sage.misc.profiler import Profiler
prof = Profiler()
prof("setup")
K = self.base_ring()
p = K.prime()
prec = K.precision_cap()
g = self.genus()
dim = 2 * g
V = VectorSpace(K, dim)
#if P or Q is Weierstrass, use the Frobenius algorithm
if self.is_weierstrass(P):
if self.is_weierstrass(Q):
return V(0)
else:
PP = None
QQ = Q
TP = None
TQ = self.frobenius(Q)
elif self.is_weierstrass(Q):
PP = P
QQ = None
TQ = None
TP = self.frobenius(P)
elif self.is_same_disc(P, Q):
return self.tiny_integrals_on_basis(P, Q)
elif algorithm == 'teichmuller':
prof("teichmuller")
PP = TP = self.teichmuller(P)
QQ = TQ = self.teichmuller(Q)
evalP, evalQ = TP, TQ
else:
prof("frobPQ")
TP = self.frobenius(P)
TQ = self.frobenius(Q)
PP, QQ = P, Q
prof("tiny integrals")
if TP is None:
P_to_TP = V(0)
else:
if TP is not None:
TPv = (TP[0]**g / TP[1]).valuation()
xTPv = TP[0].valuation()
else:
xTPv = TPv = +Infinity
if TQ is not None:
TQv = (TQ[0]**g / TQ[1]).valuation()
xTQv = TQ[0].valuation()
else:
xTQv = TQv = +Infinity
offset = (2 * g - 1) * max(TPv, TQv)
if offset == +Infinity:
offset = (2 * g - 1) * min(TPv, TQv)
if (offset > prec and (xTPv < 0 or xTQv < 0) and
(self.residue_disc(P) == self.change_ring(GF(p))(0, 1, 0)
or self.residue_disc(Q) == self.change_ring(GF(p))(0, 1, 0))):
newprec = offset + prec
K = pAdicField(p, newprec)
A = PolynomialRing(RationalField(), 'x')
f = A(self.hyperelliptic_polynomials()[0])
from sage.schemes.hyperelliptic_curves.constructor import HyperellipticCurve
self = HyperellipticCurve(f).change_ring(K)
xP = P[0]
xPv = xP.valuation()
xPnew = K(
sum(c * p**(xPv + i)
for i, c in enumerate(xP.expansion())))
PP = P = self.lift_x(xPnew)
TP = self.frobenius(P)
xQ = Q[0]
xQv = xQ.valuation()
xQnew = K(
sum(c * p**(xQv + i)
for i, c in enumerate(xQ.expansion())))
QQ = Q = self.lift_x(xQnew)
TQ = self.frobenius(Q)
V = VectorSpace(K, dim)
P_to_TP = V(self.tiny_integrals_on_basis(P, TP))
if TQ is None:
TQ_to_Q = V(0)
else:
TQ_to_Q = V(self.tiny_integrals_on_basis(TQ, Q))
prof("mw calc")
try:
M_frob, forms = self._frob_calc
except AttributeError:
M_frob, forms = self._frob_calc = monsky_washnitzer.matrix_of_frobenius_hyperelliptic(
self)
prof("eval f")
R = forms[0].base_ring()
try:
prof("eval f %s" % R)
if PP is None:
L = [-f(R(QQ[0]), R(QQ[1])) for f in forms] ##changed
elif QQ is None:
L = [f(R(PP[0]), R(PP[1])) for f in forms]
else:
L = [
f(R(PP[0]), R(PP[1])) - f(R(QQ[0]), R(QQ[1]))
for f in forms
]
except ValueError:
prof("changing rings")
forms = [f.change_ring(self.base_ring()) for f in forms]
prof("eval f %s" % self.base_ring())
if PP is None:
L = [-f(QQ[0], QQ[1]) for f in forms] ##changed
elif QQ is None:
L = [f(PP[0], PP[1]) for f in forms]
else:
L = [f(PP[0], PP[1]) - f(QQ[0], QQ[1]) for f in forms]
b = V(L)
if PP is None:
b -= TQ_to_Q
elif QQ is None:
b -= P_to_TP
elif algorithm != 'teichmuller':
b -= P_to_TP + TQ_to_Q
prof("lin alg")
M_sys = matrix(K, M_frob).transpose() - 1
TP_to_TQ = M_sys**(-1) * b
prof("done")
# print prof
if algorithm == 'teichmuller':
return P_to_TP + TP_to_TQ + TQ_to_Q
else:
return TP_to_TQ
def HyperellipticCurve_from_invariants(i,
reduced=True,
precision=None,
algorithm='default'):
r"""
Returns a hyperelliptic curve with the given Igusa-Clebsch invariants up to
scaling.
The output is a curve over the field in which the Igusa-Clebsch invariants
are given. The output curve is unique up to isomorphism over the algebraic
closure. If no such curve exists over the given field, then raise a
ValueError.
INPUT:
- ``i`` - list or tuple of length 4 containing the four Igusa-Clebsch
invariants: I2,I4,I6,I10.
- ``reduced`` - Boolean (default = True) If True, tries to reduce the
polynomial defining the hyperelliptic curve using the function
:func:`reduce_polynomial` (see the :func:`reduce_polynomial`
documentation for more details).
- ``precision`` - integer (default = None) Which precision for real and
complex numbers should the reduction use. This only affects the
reduction, not the correctness. If None, the algorithm uses the default
53 bit precision.
- ``algorithm`` - ``'default'`` or ``'magma'``. If set to ``'magma'``, uses
Magma to parameterize Mestre's conic (needs Magma to be installed).
OUTPUT:
A hyperelliptic curve object.
EXAMPLES:
Examples over the rationals::
sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856])
Traceback (most recent call last):
...
NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756.
sage: HyperellipticCurve_from_invariants([3840,414720,491028480,2437709561856],reduced = False)
Hyperelliptic Curve over Rational Field defined by y^2 = -46656*x^6 + 46656*x^5 - 19440*x^4 + 4320*x^3 - 540*x^2 + 4410*x - 1
sage: HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1])
Traceback (most recent call last):
...
NotImplementedError: Reduction of hyperelliptic curves not yet implemented. See trac #14755 and #14756.
An example over a finite field::
sage: HyperellipticCurve_from_invariants([GF(13)(1),3,7,5])
Hyperelliptic Curve over Finite Field of size 13 defined by y^2 = 8*x^5 + 5*x^4 + 5*x^2 + 9*x + 3
An example over a number field::
sage: K = QuadraticField(353, 'a')
sage: H = HyperellipticCurve_from_invariants([21, 225/64, 22941/512, 1], reduced = false)
sage: f = K['x'](H.hyperelliptic_polynomials()[0])
If the Mestre Conic defined by the Igusa-Clebsch invariants has no rational
points, then there exists no hyperelliptic curve over the base field with
the given invariants.::
sage: HyperellipticCurve_from_invariants([1,2,3,4])
Traceback (most recent call last):
...
ValueError: No such curve exists over Rational Field as there are no rational points on Projective Conic Curve over Rational Field defined by -2572155000*u^2 - 317736000*u*v + 1250755459200*v^2 + 2501510918400*u*w + 39276887040*v*w + 2736219686912*w^2
Mestre's algorithm only works for generic curves of genus two, so another
algorithm is needed for those curves with extra automorphism. See also
:trac:`12199`::
sage: P.<x> = QQ[]
sage: C = HyperellipticCurve(x^6+1)
sage: i = C.igusa_clebsch_invariants()
sage: HyperellipticCurve_from_invariants(i)
Traceback (most recent call last):
...
TypeError: F (=0) must have degree 2
Igusa-Clebsch invariants also only work over fields of characteristic
different from 2, 3, and 5, so another algorithm will be needed for fields
of those characteristics. See also :trac:`12200`::
sage: P.<x> = GF(3)[]
sage: HyperellipticCurve(x^6+x+1).igusa_clebsch_invariants()
Traceback (most recent call last):
...
NotImplementedError: Invariants of binary sextics/genus 2 hyperelliptic curves not implemented in characteristics 2, 3, and 5
sage: HyperellipticCurve_from_invariants([GF(5)(1),1,0,1])
Traceback (most recent call last):
...
ZeroDivisionError: Inverse does not exist.
ALGORITHM:
This is Mestre's algorithm [M1991]_. Our implementation is based on the
formulae on page 957 of [LY2001]_, cross-referenced with [W1999]_ to
correct typos.
First construct Mestre's conic using the :func:`Mestre_conic` function.
Parametrize the conic if possible.
Let `f_1, f_2, f_3` be the three coordinates of the parametrization of the
conic by the projective line, and change them into one variable by letting
`F_i = f_i(t, 1)`. Note that each `F_i` has degree at most 2.
Then construct a sextic polynomial
`f = \sum_{0<=i,j,k<=3}{c_{ijk}*F_i*F_j*F_k}`,
where `c_{ijk}` are defined as rational functions in the invariants
(see the source code for detailed formulae for `c_{ijk}`).
The output is the hyperelliptic curve `y^2 = f`.
REFERENCES:
.. [LY2001] \K. Lauter and T. Yang, "Computing genus 2 curves from
invariants on the Hilbert moduli space", Journal of Number Theory 131
(2011), pages 936 - 958
.. [M1991] \J.-F. Mestre, "Construction de courbes de genre 2 a partir de
leurs modules", in Effective methods in algebraic geometry
(Castiglioncello, 1990), volume 94 of Progr. Math., pages 313 - 334
.. [W1999] \P. van Wamelen, Pari-GP code, section "thecubic"
https://www.math.lsu.edu/~wamelen/Genus2/FindCurve/igusa2curve.gp
"""
from sage.structure.sequence import Sequence
i = Sequence(i)
k = i.universe()
try:
k = k.fraction_field()
except (TypeError, AttributeError, NotImplementedError):
pass
MConic, x, y, z = Mestre_conic(i, xyz=True)
if k.is_finite():
reduced = False
t = k['t'].gen()
if algorithm == 'magma':
from sage.interfaces.all import magma
from sage.misc.sage_eval import sage_eval
if MConic.has_rational_point(algorithm='magma'):
parametrization = [l.replace('$.1', 't').replace('$.2', 'u') \
for l in str(magma(MConic).Parametrization()).splitlines()[4:7]]
[F1, F2, F3] = [sage_eval(p, locals={'t':t,'u':1,'a':k.gen()}) \
for p in parametrization]
else:
raise ValueError("No such curve exists over %s as there are no " \
"rational points on %s" % (k, MConic))
else:
if MConic.has_rational_point():
parametrization = MConic.parametrization(morphism=False)[0]
[F1, F2, F3] = [p(t, 1) for p in parametrization]
else:
raise ValueError("No such curve exists over %s as there are no " \
"rational points on %s" % (k, MConic))
# setting the cijk from Mestre's algorithm
c111 = 12 * x * y - 2 * y / 3 - 4 * z
c112 = -18 * x**3 - 12 * x * y - 36 * y**2 - 2 * z
c113 = -9 * x**3 - 36 * x**2 * y - 4 * x * y - 6 * x * z - 18 * y**2
c122 = c113
c123 = -54 * x**4 - 36 * x**2 * y - 36 * x * y**2 - 6 * x * z - 4 * y**2 - 24 * y * z
c133 = -27*x**4/2 - 72*x**3*y - 6*x**2*y - 9*x**2*z - 39*x*y**2 - \
36*y**3 - 2*y*z
c222 = -27 * x**4 - 18 * x**2 * y - 6 * x * y**2 - 8 * y**2 / 3 + 2 * y * z
c223 = 9 * x**3 * y - 27 * x**2 * z + 6 * x * y**2 + 18 * y**3 - 8 * y * z
c233 = -81 * x**5 / 2 - 27 * x**3 * y - 9 * x**2 * y**2 - 4 * x * y**2 + 3 * x * y * z - 6 * z**2
c333 = 27*x**4*y/2 - 27*x**3*z/2 + 9*x**2*y**2 + 3*x*y**3 - 6*x*y*z + \
4*y**3/3 - 10*y**2*z
# writing out the hyperelliptic curve polynomial
f = c111*F1**3 + c112*F1**2*F2 + c113*F1**2*F3 + c122*F1*F2**2 + \
c123*F1*F2*F3 + c133*F1*F3**2 + c222*F2**3 + c223*F2**2*F3 + \
c233*F2*F3**2 + c333*F3**3
try:
f = f * f.denominator() # clear the denominator
except (AttributeError, TypeError):
pass
if reduced:
raise NotImplementedError("Reduction of hyperelliptic curves not " \
"yet implemented. " \
"See trac #14755 and #14756.")
return HyperellipticCurve(f)
