def _tate(self, proof = None, globally = False):
r"""
Tate's algorithm for an elliptic curve over a number field.
Computes both local reduction data at a prime ideal and a
local minimal model.
The model is not required to be integral on input. If `P` is
principal, uses a generator as uniformizer, so it will not
affect integrality or minimality at other primes. If `P` is not
principal, the minimal model returned will preserve
integrality at other primes, but not minimality.
The optional argument globally, when set to True, tells the algorithm to use the generator of the prime ideal if it is principal. Otherwise just any uniformizer will be used.
.. note::
Called only by ``EllipticCurveLocalData.__init__()``.
OUTPUT:
(tuple) ``(Emin, p, val_disc, fp, KS, cp)`` where:
- ``Emin`` (EllipticCurve) is a model (integral and) minimal at P
- ``p`` (int) is the residue characteristic
- ``val_disc`` (int) is the valuation of the local minimal discriminant
- ``fp`` (int) is the valuation of the conductor
- ``KS`` (string) is the Kodaira symbol
- ``cp`` (int) is the Tamagawa number
EXAMPLES (this raised a type error in sage prior to 4.4.4, see :trac:`7930`) ::
sage: E = EllipticCurve('99d1')
sage: R.<X> = QQ[]
sage: K.<t> = NumberField(X^3 + X^2 - 2*X - 1)
sage: L.<s> = NumberField(X^3 + X^2 - 36*X - 4)
sage: EK = E.base_extend(K)
sage: toK = EK.torsion_order()
sage: da = EK.local_data() # indirect doctest
sage: EL = E.base_extend(L)
sage: da = EL.local_data() # indirect doctest
EXAMPLES:
The following example shows that the bug at :trac:`9324` is fixed::
sage: K.<a> = NumberField(x^2-x+6)
sage: E = EllipticCurve([0,0,0,-53160*a-43995,-5067640*a+19402006])
sage: E.conductor() # indirect doctest
Fractional ideal (18, 6*a)
The following example shows that the bug at :trac:`9417` is fixed::
sage: K.<a> = NumberField(x^2+18*x+1)
sage: E = EllipticCurve(K, [0, -36, 0, 320, 0])
sage: E.tamagawa_number(K.ideal(2))
4
This is to show that the bug :trac: `11630` is fixed. (The computation of the class group would produce a warning)::
sage: K.<t> = NumberField(x^7-2*x+177)
sage: E = EllipticCurve([0,1,0,t,t])
sage: P = K.ideal(2,t^3 + t + 1)
sage: E.local_data(P).kodaira_symbol()
II
"""
E = self._curve
P = self._prime
K = E.base_ring()
OK = K.maximal_order()
t = verbose("Running Tate's algorithm with P = %s"%P, level=1)
F = OK.residue_field(P)
p = F.characteristic()
# In case P is not principal we mostly use a uniformiser which
# is globally integral (with positive valuation at some other
# primes); for this to work, it is essential that we can
# reduce (mod P) elements of K which are not integral (but are
# P-integral). However, if the model is non-minimal and we
# end up dividing a_i by pi^i then at that point we use a
# uniformiser pi which has non-positive valuation at all other
# primes, so that we can divide by it without losing
# integrality at other primes.
if globally:
principal_flag = P.is_principal()
else:
principal_flag = False
if (K is QQ) or principal_flag :
pi = P.gens_reduced()[0]
verbose("P is principal, generator pi = %s"%pi, t, 1)
else:
pi = K.uniformizer(P, 'positive')
verbose("uniformizer pi = %s"%pi, t, 1)
pi2 = pi*pi; pi3 = pi*pi2; pi4 = pi*pi3
pi_neg = None
prime = pi if K is QQ else P
pval = lambda x: x.valuation(prime)
pdiv = lambda x: x.is_zero() or pval(x) > 0
# Since ResidueField is cached in a way that
# does not care much about embeddings of number
# fields, it can happen that F.p.ring() is different
# from K. This is a problem: If F.p.ring() has no
# embedding but K has, then there is no coercion
# from F.p.ring().maximal_order() to K. But it is
# no problem to do an explicit conversion in that
# case (Simon King, trac ticket #8800).
from sage.categories.pushout import pushout, CoercionException
try:
if hasattr(F.p.ring(), 'maximal_order'): # it is not ZZ
_tmp_ = pushout(F.p.ring().maximal_order(),K)
pinv = lambda x: F.lift(~F(x))
proot = lambda x,e: F.lift(F(x).nth_root(e, extend = False, all = True)[0])
preduce = lambda x: F.lift(F(x))
except CoercionException: # the pushout does not exist, we need conversion
pinv = lambda x: K(F.lift(~F(x)))
proot = lambda x,e: K(F.lift(F(x).nth_root(e, extend = False, all = True)[0]))
preduce = lambda x: K(F.lift(F(x)))
def _pquadroots(a, b, c):
r"""
Local function returning True iff `ax^2 + bx + c` has roots modulo `P`
"""
(a, b, c) = (F(a), F(b), F(c))
if a == 0:
return (b != 0) or (c == 0)
elif p == 2:
return len(PolynomialRing(F, "x")([c,b,a]).roots()) > 0
else:
return (b**2 - 4*a*c).is_square()
def _pcubicroots(b, c, d):
r"""
Local function returning the number of roots of `x^3 +
b*x^2 + c*x + d` modulo `P`, counting multiplicities
"""
return sum([rr[1] for rr in PolynomialRing(F, 'x')([F(d), F(c), F(b), F(1)]).roots()],0)
if p == 2:
halfmodp = OK(Integer(0))
else:
halfmodp = pinv(Integer(2))
A = E.a_invariants()
A = [0, A[0], A[1], A[2], A[3], 0, A[4]]
indices = [1,2,3,4,6]
if min([pval(a) for a in A if a != 0]) < 0:
verbose("Non-integral model at P: valuations are %s; making integral"%([pval(a) for a in A if a != 0]), t, 1)
e = 0
for i in range(7):
if A[i] != 0:
e = max(e, (-pval(A[i])/i).ceil())
pie = pi**e
for i in range(7):
if A[i] != 0:
A[i] *= pie**i
verbose("P-integral model is %s, with valuations %s"%([A[i] for i in indices], [pval(A[i]) for i in indices]), t, 1)
split = None # only relevant for multiplicative reduction
(a1, a2, a3, a4, a6) = (A[1], A[2], A[3], A[4], A[6])
while True:
C = EllipticCurve([a1, a2, a3, a4, a6]);
(b2, b4, b6, b8) = C.b_invariants()
(c4, c6) = C.c_invariants()
delta = C.discriminant()
val_disc = pval(delta)
if val_disc == 0:
## Good reduction already
cp = 1
fp = 0
KS = KodairaSymbol("I0")
break #return
# Otherwise, we change coordinates so that p | a3, a4, a6
if p == 2:
if pdiv(b2):
r = proot(a4, 2)
t = proot(((r + a2)*r + a4)*r + a6, 2)
else:
temp = pinv(a1)
r = temp * a3
t = temp * (a4 + r*r)
elif p == 3:
if pdiv(b2):
r = proot(-b6, 3)
else:
r = -pinv(b2) * b4
t = a1 * r + a3
else:
if pdiv(c4):
r = -pinv(12) * b2
else:
r = -pinv(12*c4) * (c6 + b2 * c4)
t = -halfmodp * (a1 * r + a3)
r = preduce(r)
t = preduce(t)
verbose("Before first transform C = %s"%C)
verbose("[a1,a2,a3,a4,a6] = %s"%([a1, a2, a3, a4, a6]))
C = C.rst_transform(r, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
if min([pval(a) for a in (a1, a2, a3, a4, a6) if a != 0]) < 0:
raise RuntimeError("Non-integral model after first transform!")
verbose("After first transform %s\n, [a1,a2,a3,a4,a6] = %s\n, valuations = %s"%([r, 0, t], [a1, a2, a3, a4, a6], [pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)]), t, 2)
if pval(a3) == 0:
raise RuntimeError("p does not divide a3 after first transform!")
if pval(a4) == 0:
raise RuntimeError("p does not divide a4 after first transform!")
if pval(a6) == 0:
raise RuntimeError("p does not divide a6 after first transform!")
# Now we test for Types In, II, III, IV
# NB the c invariants never change.
if not pdiv(c4):
# Multiplicative reduction: Type In (n = val_disc)
split = False
if _pquadroots(1, a1, -a2):
cp = val_disc
split = True
elif Integer(2).divides(val_disc):
cp = 2
else:
cp = 1
KS = KodairaSymbol("I%s"%val_disc)
fp = 1
break #return
# Additive reduction
if pval(a6) < 2:
## Type II
KS = KodairaSymbol("II")
fp = val_disc
cp = 1
break #return
if pval(b8) < 3:
## Type III
KS = KodairaSymbol("III")
fp = val_disc - 1
cp = 2
break #return
if pval(b6) < 3:
## Type IV
cp = 1
a3t = preduce(a3/pi)
a6t = preduce(a6/pi2)
if _pquadroots(1, a3t, -a6t): cp = 3
KS = KodairaSymbol("IV")
fp = val_disc - 2
break #return
# If our curve is none of these types, we change coords so that
# p | a1, a2; p^2 | a3, a4; p^3 | a6
if p == 2:
s = proot(a2, 2) # so s^2=a2 (mod pi)
t = pi*proot(a6/pi2, 2) # so t^2=a6 (mod pi^3)
elif p == 3:
s = a1 # so a1'=2s+a1=3a1=0 (mod pi)
t = a3 # so a3'=2t+a3=3a3=0 (mod pi^2)
else:
s = -a1*halfmodp # so a1'=2s+a1=0 (mod pi)
t = -a3*halfmodp # so a3'=2t+a3=0 (mod pi^2)
C = C.rst_transform(0, s, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
verbose("After second transform %s\n[a1, a2, a3, a4, a6] = %s\nValuations: %s"%([0, s, t], [a1,a2,a3,a4,a6],[pval(a1),pval(a2),pval(a3),pval(a4),pval(a6)]), t, 2)
if pval(a1) == 0:
raise RuntimeError("p does not divide a1 after second transform!")
if pval(a2) == 0:
raise RuntimeError("p does not divide a2 after second transform!")
if pval(a3) < 2:
raise RuntimeError("p^2 does not divide a3 after second transform!")
if pval(a4) < 2:
raise RuntimeError("p^2 does not divide a4 after second transform!")
if pval(a6) < 3:
raise RuntimeError("p^3 does not divide a6 after second transform!")
if min(pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)) < 0:
raise RuntimeError("Non-integral model after second transform!")
# Analyze roots of the cubic T^3 + bT^2 + cT + d = 0 mod P, where
# b = a2/p, c = a4/p^2, d = a6/p^3
b = preduce(a2/pi)
c = preduce(a4/pi2)
d = preduce(a6/pi3)
bb = b*b
cc = c*c
bc = b*c
w = 27*d*d - bb*cc + 4*b*bb*d - 18*bc*d + 4*c*cc
x = 3*c - bb
if pdiv(w):
if pdiv(x):
sw = 3
else:
sw = 2
else:
sw = 1
verbose("Analyzing roots of cubic T^3 + %s*T^2 + %s*T + %s, case %s"%(b, c, d, sw), t, 1)
if sw == 1:
## Three distinct roots - Type I*0
verbose("Distinct roots", t, 1)
KS = KodairaSymbol("I0*")
cp = 1 + _pcubicroots(b, c, d)
fp = val_disc - 4
break #return
elif sw == 2:
## One double root - Type I*m for some m
verbose("One double root", t, 1)
## Change coords so that the double root is T = 0 mod p
if p == 2:
r = proot(c, 2)
elif p == 3:
r = c * pinv(b)
else:
r = (bc - 9*d)*pinv(2*x)
r = pi * preduce(r)
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
# The rest of this branch is just to compute cp, fp, KS.
# We use pi to keep transforms integral.
ix = 3; iy = 3; mx = pi2; my = mx
while True:
a2t = preduce(a2 / pi)
a3t = preduce(a3 / my)
a4t = preduce(a4 / (pi*mx))
a6t = preduce(a6 / (mx*my))
if pdiv(a3t*a3t + 4*a6t):
if p == 2:
t = my*proot(a6t, 2)
else:
t = my*preduce(-a3t*halfmodp)
C = C.rst_transform(0, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
my *= pi
iy += 1
a2t = preduce(a2 / pi)
a3t = preduce(a3/my)
a4t = preduce(a4/(pi*mx))
a6t = preduce(a6/(mx*my))
if pdiv(a4t*a4t - 4*a6t*a2t):
if p == 2:
r = mx*proot(a6t*pinv(a2t), 2)
else:
r = mx*preduce(-a4t*pinv(2*a2t))
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
mx *= pi
ix += 1 # and stay in loop
else:
if _pquadroots(a2t, a4t, a6t):
cp = 4
else:
cp = 2
break # exit loop
else:
if _pquadroots(1, a3t, -a6t):
cp = 4
else:
cp = 2
break
KS = KodairaSymbol("I%s*"%(ix+iy-5))
fp = val_disc - ix - iy + 1
break #return
else: # sw == 3
## The cubic has a triple root
verbose("Triple root", t, 1)
## First we change coordinates so that T = 0 mod p
if p == 2:
r = b
elif p == 3:
r = proot(-d, 3)
else:
r = -b * pinv(3)
r = pi*preduce(r)
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
verbose("After third transform %s\n[a1,a2,a3,a4,a6] = %s\nValuations: %s"%([r,0,0],[a1,a2,a3,a4,a6],[pval(ai) for ai in [a1,a2,a3,a4,a6]]), t, 2)
if min(pval(ai) for ai in [a1,a2,a3,a4,a6]) < 0:
raise RuntimeError("Non-integral model after third transform!")
if pval(a2) < 2 or pval(a4) < 3 or pval(a6) < 4:
raise RuntimeError("Cubic after transform does not have a triple root at 0")
a3t = preduce(a3/pi2)
a6t = preduce(a6/pi4)
# We test for Type IV*
if not pdiv(a3t*a3t + 4*a6t):
cp = 3 if _pquadroots(1, a3t, -a6t) else 1
KS = KodairaSymbol("IV*")
fp = val_disc - 6
break #return
# Now change coordinates so that p^3|a3, p^5|a6
if p==2:
t = -pi2*proot(a6t, 2)
else:
t = pi2*preduce(-a3t*halfmodp)
C = C.rst_transform(0, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
# We test for types III* and II*
if pval(a4) < 4:
## Type III*
KS = KodairaSymbol("III*")
fp = val_disc - 7
cp = 2
break #return
if pval(a6) < 6:
## Type II*
KS = KodairaSymbol("II*")
fp = val_disc - 8
cp = 1
break #return
if pi_neg is None:
if principal_flag:
pi_neg = pi
else:
pi_neg = K.uniformizer(P, 'negative')
pi_neg2 = pi_neg*pi_neg
pi_neg3 = pi_neg*pi_neg2
pi_neg4 = pi_neg*pi_neg3
pi_neg6 = pi_neg4*pi_neg2
a1 /= pi_neg
a2 /= pi_neg2
a3 /= pi_neg3
a4 /= pi_neg4
a6 /= pi_neg6
verbose("Non-minimal equation, dividing out...\nNew model is %s"%([a1, a2, a3, a4, a6]), t, 1)
return (C, p, val_disc, fp, KS, cp, split)
def descend_to(self, K, f=None):
r"""
Given a subfield `K` and an elliptic curve self defined over a field `L`,
this function determines whether there exists an elliptic curve over `K`
which is isomorphic over `L` to self. If one exists, it finds it.
INPUT:
- `K` -- a subfield of the base field of self.
- `f` -- an embedding of `K` into the base field of self.
OUTPUT:
Either an elliptic curve defined over `K` which is isomorphic to self
or None if no such curve exists.
.. NOTE::
This only works over number fields and QQ.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.descend_to(ZZ)
Traceback (most recent call last):
...
TypeError: Input must be a field.
::
sage: F.<b> = QuadraticField(23)
sage: G.<a> = F.extension(x^3+5)
sage: E = EllipticCurve(j=1728*b).change_ring(G)
sage: E.descend_to(F)
Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23
::
sage: L.<a> = NumberField(x^4 - 7)
sage: K.<b> = NumberField(x^2 - 7)
sage: E = EllipticCurve([a^6,0])
sage: E.descend_to(K)
Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7
::
sage: K.<a> = QuadraticField(17)
sage: E = EllipticCurve(j = 2*a)
sage: print E.descend_to(QQ)
None
"""
if not K.is_field():
raise TypeError, "Input must be a field."
if self.base_field()==K:
return self
j = self.j_invariant()
from sage.rings.all import QQ
if K == QQ:
f = QQ.embeddings(self.base_field())[0]
if j in QQ:
jbase = QQ(j)
else:
return None
elif f == None:
embeddings = K.embeddings(self.base_field())
if len(embeddings) == 0:
raise TypeError, "Input must be a subfield of the base field of the curve."
for g in embeddings:
try:
jbase = g.preimage(j)
f = g
break
except StandardError:
pass
if f == None:
return None
else:
try:
jbase = f.preimage(j)
except StandardError:
return None
E = EllipticCurve(j=jbase)
E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()])
if jbase==0:
d = self.is_sextic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.sextic_twist(d)
elif jbase==1728:
d = self.is_quartic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quartic_twist(d)
else:
d = self.is_quadratic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quadratic_twist(d)
if Etwist.is_isomorphic(self):
try:
Eout = EllipticCurve(K, [f.preimage(a) for a in Etwist.a_invariants()])
except StandardError:
return None
else:
return Eout
def __init__(self, E=None, urst=None, F=None):
r"""
Constructor for WeierstrassIsomorphism class,
INPUT:
- ``E`` -- an EllipticCurve, or None (see below).
- ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below).
- ``F`` -- an EllipticCurve, or None (see below).
Given two Elliptic Curves ``E`` and ``F`` (represented by
Weierstrass models as usual), and a transformation ``urst``
from ``E`` to ``F``, construct an isomorphism from ``E`` to
``F``. An exception is raised if ``urst(E)!=F``. At most one
of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then
``F`` is constructed as ``urst(E)``. If ``E==None`` then
``E`` is constructed as ``urst^-1(F)``. If ``urst==None``
then an isomorphism from ``E`` to ``F`` is constructed if
possible, and an exception is raised if they are not
isomorphic. Otherwise ``urst`` can be a tuple of length 4 or
a object of type ``baseWI``.
Users will not usually need to use this class directly, but instead use
methods such as ``isomorphism`` of elliptic curves.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4))
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field
Via: (u,r,s,t) = (-1, 2, 3, 4)
sage: E=EllipticCurve([0,1,2,3,4])
sage: F=EllipticCurve(E.cremona_label())
sage: WeierstrassIsomorphism(E,None,F)
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 0, 0, -1)
sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F)
sage: w._domain_curve==E
True
"""
from ell_generic import is_EllipticCurve
if E != None:
if not is_EllipticCurve(E):
raise ValueError, "First argument must be an elliptic curve or None"
if F != None:
if not is_EllipticCurve(F):
raise ValueError, "Third argument must be an elliptic curve or None"
if urst != None:
if len(urst) != 4:
raise ValueError, "Second argument must be [u,r,s,t] or None"
if len([par for par in [E, urst, F] if par != None]) < 2:
raise ValueError, "At most 1 argument can be None"
if F == None: # easy case
baseWI.__init__(self, *urst)
F = EllipticCurve(baseWI.__call__(self, list(E.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if E == None: # easy case in reverse
baseWI.__init__(self, *urst)
inv_urst = baseWI.__invert__(self)
E = EllipticCurve(baseWI.__call__(inv_urst,
list(F.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if urst == None: # try to construct the morphism
urst = isomorphisms(E, F, True)
if urst == None:
raise ValueError, "Elliptic curves not isomorphic."
baseWI.__init__(self, *urst)
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
# none of the parameters is None:
baseWI.__init__(self, *urst)
if F != EllipticCurve(baseWI.__call__(self, list(E.a_invariants()))):
raise ValueError, "second argument is not an isomorphism from first argument to third argument"
else:
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
def __init__(self, E=None, urst=None, F=None):
r"""
Constructor for WeierstrassIsomorphism class,
INPUT:
- ``E`` -- an EllipticCurve, or None (see below).
- ``urst`` -- a 4-tuple `(u,r,s,t)`, or None (see below).
- ``F`` -- an EllipticCurve, or None (see below).
Given two Elliptic Curves ``E`` and ``F`` (represented by
Weierstrass models as usual), and a transformation ``urst``
from ``E`` to ``F``, construct an isomorphism from ``E`` to
``F``. An exception is raised if ``urst(E)!=F``. At most one
of ``E``, ``F``, ``urst`` can be None. If ``F==None`` then
``F`` is constructed as ``urst(E)``. If ``E==None`` then
``E`` is constructed as ``urst^-1(F)``. If ``urst==None``
then an isomorphism from ``E`` to ``F`` is constructed if
possible, and an exception is raised if they are not
isomorphic. Otherwise ``urst`` can be a tuple of length 4 or
a object of type ``baseWI``.
Users will not usually need to use this class directly, but instead use
methods such as ``isomorphism`` of elliptic curves.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.weierstrass_morphism import *
sage: WeierstrassIsomorphism(EllipticCurve([0,1,2,3,4]),(-1,2,3,4))
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 - 6*x*y - 10*y = x^3 - 2*x^2 - 11*x - 2 over Rational Field
Via: (u,r,s,t) = (-1, 2, 3, 4)
sage: E=EllipticCurve([0,1,2,3,4])
sage: F=EllipticCurve(E.cremona_label())
sage: WeierstrassIsomorphism(E,None,F)
Generic morphism:
From: Abelian group of points on Elliptic Curve defined by y^2 + 2*y = x^3 + x^2 + 3*x + 4 over Rational Field
To: Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x^2 + 3*x + 5 over Rational Field
Via: (u,r,s,t) = (1, 0, 0, -1)
sage: w=WeierstrassIsomorphism(None,(1,0,0,-1),F)
sage: w._domain_curve==E
True
"""
from ell_generic import is_EllipticCurve
if E!=None:
if not is_EllipticCurve(E):
raise ValueError("First argument must be an elliptic curve or None")
if F!=None:
if not is_EllipticCurve(F):
raise ValueError("Third argument must be an elliptic curve or None")
if urst!=None:
if len(urst)!=4:
raise ValueError("Second argument must be [u,r,s,t] or None")
if len([par for par in [E,urst,F] if par!=None])<2:
raise ValueError("At most 1 argument can be None")
if F==None: # easy case
baseWI.__init__(self,*urst)
F=EllipticCurve(baseWI.__call__(self,list(E.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if E==None: # easy case in reverse
baseWI.__init__(self,*urst)
inv_urst=baseWI.__invert__(self)
E=EllipticCurve(baseWI.__call__(inv_urst,list(F.a_invariants())))
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
if urst==None: # try to construct the morphism
urst=isomorphisms(E,F,True)
if urst==None:
raise ValueError("Elliptic curves not isomorphic.")
baseWI.__init__(self, *urst)
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
# none of the parameters is None:
baseWI.__init__(self,*urst)
if F!=EllipticCurve(baseWI.__call__(self,list(E.a_invariants()))):
raise ValueError("second argument is not an isomorphism from first argument to third argument")
else:
Morphism.__init__(self, Hom(E(0).parent(), F(0).parent()))
self._domain_curve = E
self._codomain_curve = F
return
def _tate(self, proof=None):
r"""
Tate's algorithm for an elliptic curve over a number field.
Computes both local reduction data at a prime ideal and a
local minimal model.
The model is not required to be integral on input. If `P` is
principal, uses a generator as uniformizer, so it will not
affect integrality or minimality at other primes. If `P` is not
principal, the minimal model returned will preserve
integrality at other primes, but not minimality.
.. note::
Called only by ``EllipticCurveLocalData.__init__()``.
OUTPUT:
(tuple) ``(Emin, p, val_disc, fp, KS, cp)`` where:
- ``Emin`` (EllipticCurve) is a model (integral and) minimal at P
- ``p`` (int) is the residue characteristic
- ``val_disc`` (int) is the valuation of the local minimal discriminant
- ``fp`` (int) is the valuation of the conductor
- ``KS`` (string) is the Kodaira symbol
- ``cp`` (int) is the Tamagawa number
EXAMPLES (this raised a type error in sage prior to 4.4.4, see ticket #7930) ::
sage: E = EllipticCurve('99d1')
sage: R.<X> = QQ[]
sage: K.<t> = NumberField(X^3 + X^2 - 2*X - 1)
sage: L.<s> = NumberField(X^3 + X^2 - 36*X - 4)
sage: EK = E.base_extend(K)
sage: toK = EK.torsion_order()
sage: da = EK.local_data() # indirect doctest
sage: EL = E.base_extend(L)
sage: da = EL.local_data() # indirect doctest
EXAMPLES:
The following example shows that the bug at #9324 is fixed::
sage: K.<a> = NumberField(x^2-x+6)
sage: E = EllipticCurve([0,0,0,-53160*a-43995,-5067640*a+19402006])
sage: E.conductor() # indirect doctest
Fractional ideal (18, 6*a)
The following example shows that the bug at #9417 is fixed::
sage: K.<a> = NumberField(x^2+18*x+1)
sage: E = EllipticCurve(K, [0, -36, 0, 320, 0])
sage: E.tamagawa_number(K.ideal(2))
4
"""
E = self._curve
P = self._prime
K = E.base_ring()
OK = K.maximal_order()
t = verbose("Running Tate's algorithm with P = %s" % P, level=1)
F = OK.residue_field(P)
p = F.characteristic()
# In case P is not principal we mostly use a uniformiser which
# is globally integral (with positive valuation at some other
# primes); for this to work, it is essential that we can
# reduce (mod P) elements of K which are not integral (but are
# P-integral). However, if the model is non-minimal and we
# end up dividing a_i by pi^i then at that point we use a
# uniformiser pi which has non-positive valuation at all other
# primes, so that we can divide by it without losing
# integrality at other primes.
principal_flag = P.is_principal()
if principal_flag:
pi = P.gens_reduced()[0]
verbose("P is principal, generator pi = %s" % pi, t, 1)
else:
pi = K.uniformizer(P, 'positive')
verbose("P is not principal, uniformizer pi = %s" % pi, t, 1)
pi2 = pi * pi
pi3 = pi * pi2
pi4 = pi * pi3
pi_neg = None
prime = pi if K is QQ else P
pval = lambda x: x.valuation(prime)
pdiv = lambda x: x.is_zero() or pval(x) > 0
# Since ResidueField is cached in a way that
# does not care much about embeddings of number
# fields, it can happen that F.p.ring() is different
# from K. This is a problem: If F.p.ring() has no
# embedding but K has, then there is no coercion
# from F.p.ring().maximal_order() to K. But it is
# no problem to do an explicit conversion in that
# case (Simon King, trac ticket #8800).
from sage.categories.pushout import pushout, CoercionException
try:
if hasattr(F.p.ring(), 'maximal_order'): # it is not ZZ
_tmp_ = pushout(F.p.ring().maximal_order(), K)
pinv = lambda x: F.lift(~F(x))
proot = lambda x, e: F.lift(
F(x).nth_root(e, extend=False, all=True)[0])
preduce = lambda x: F.lift(F(x))
except CoercionException: # the pushout does not exist, we need conversion
pinv = lambda x: K(F.lift(~F(x)))
proot = lambda x, e: K(
F.lift(F(x).nth_root(e, extend=False, all=True)[0]))
preduce = lambda x: K(F.lift(F(x)))
def _pquadroots(a, b, c):
r"""
Local function returning True iff `ax^2 + bx + c` has roots modulo `P`
"""
(a, b, c) = (F(a), F(b), F(c))
if a == 0:
return (b != 0) or (c == 0)
elif p == 2:
return len(PolynomialRing(F, "x")([c, b, a]).roots()) > 0
else:
return (b**2 - 4 * a * c).is_square()
def _pcubicroots(b, c, d):
r"""
Local function returning the number of roots of `x^3 +
b*x^2 + c*x + d` modulo `P`, counting multiplicities
"""
return sum([
rr[1] for rr in PolynomialRing(F, 'x')
([F(d), F(c), F(b), F(1)]).roots()
], 0)
if p == 2:
halfmodp = OK(Integer(0))
else:
halfmodp = pinv(Integer(2))
A = E.a_invariants()
A = [0, A[0], A[1], A[2], A[3], 0, A[4]]
indices = [1, 2, 3, 4, 6]
if min([pval(a) for a in A if a != 0]) < 0:
verbose(
"Non-integral model at P: valuations are %s; making integral" %
([pval(a) for a in A if a != 0]), t, 1)
e = 0
for i in range(7):
if A[i] != 0:
e = max(e, (-pval(A[i]) / i).ceil())
pie = pi**e
for i in range(7):
if A[i] != 0:
A[i] *= pie**i
verbose(
"P-integral model is %s, with valuations %s" %
([A[i] for i in indices], [pval(A[i]) for i in indices]), t, 1)
split = None # only relevant for multiplicative reduction
(a1, a2, a3, a4, a6) = (A[1], A[2], A[3], A[4], A[6])
while True:
C = EllipticCurve([a1, a2, a3, a4, a6])
(b2, b4, b6, b8) = C.b_invariants()
(c4, c6) = C.c_invariants()
delta = C.discriminant()
val_disc = pval(delta)
if val_disc == 0:
## Good reduction already
cp = 1
fp = 0
KS = KodairaSymbol("I0")
break #return
# Otherwise, we change coordinates so that p | a3, a4, a6
if p == 2:
if pdiv(b2):
r = proot(a4, 2)
t = proot(((r + a2) * r + a4) * r + a6, 2)
else:
temp = pinv(a1)
r = temp * a3
t = temp * (a4 + r * r)
elif p == 3:
if pdiv(b2):
r = proot(-b6, 3)
else:
r = -pinv(b2) * b4
t = a1 * r + a3
else:
if pdiv(c4):
r = -pinv(12) * b2
else:
r = -pinv(12 * c4) * (c6 + b2 * c4)
t = -halfmodp * (a1 * r + a3)
r = preduce(r)
t = preduce(t)
verbose("Before first transform C = %s" % C)
verbose("[a1,a2,a3,a4,a6] = %s" % ([a1, a2, a3, a4, a6]))
C = C.rst_transform(r, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
if min([pval(a) for a in (a1, a2, a3, a4, a6) if a != 0]) < 0:
raise RuntimeError, "Non-integral model after first transform!"
verbose(
"After first transform %s\n, [a1,a2,a3,a4,a6] = %s\n, valuations = %s"
% ([r, 0, t], [a1, a2, a3, a4, a6],
[pval(a1), pval(a2),
pval(a3), pval(a4),
pval(a6)]), t, 2)
if pval(a3) == 0:
raise RuntimeError, "p does not divide a3 after first transform!"
if pval(a4) == 0:
raise RuntimeError, "p does not divide a4 after first transform!"
if pval(a6) == 0:
raise RuntimeError, "p does not divide a6 after first transform!"
# Now we test for Types In, II, III, IV
# NB the c invariants never change.
if not pdiv(c4):
# Multiplicative reduction: Type In (n = val_disc)
split = False
if _pquadroots(1, a1, -a2):
cp = val_disc
split = True
elif Integer(2).divides(val_disc):
cp = 2
else:
cp = 1
KS = KodairaSymbol("I%s" % val_disc)
fp = 1
break #return
# Additive reduction
if pval(a6) < 2:
## Type II
KS = KodairaSymbol("II")
fp = val_disc
cp = 1
break #return
if pval(b8) < 3:
## Type III
KS = KodairaSymbol("III")
fp = val_disc - 1
cp = 2
break #return
if pval(b6) < 3:
## Type IV
cp = 1
a3t = preduce(a3 / pi)
a6t = preduce(a6 / pi2)
if _pquadroots(1, a3t, -a6t): cp = 3
KS = KodairaSymbol("IV")
fp = val_disc - 2
break #return
# If our curve is none of these types, we change coords so that
# p | a1, a2; p^2 | a3, a4; p^3 | a6
if p == 2:
s = proot(a2, 2) # so s^2=a2 (mod pi)
t = pi * proot(a6 / pi2, 2) # so t^2=a6 (mod pi^3)
elif p == 3:
s = a1 # so a1'=2s+a1=3a1=0 (mod pi)
t = a3 # so a3'=2t+a3=3a3=0 (mod pi^2)
else:
s = -a1 * halfmodp # so a1'=2s+a1=0 (mod pi)
t = -a3 * halfmodp # so a3'=2t+a3=0 (mod pi^2)
C = C.rst_transform(0, s, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
verbose(
"After second transform %s\n[a1, a2, a3, a4, a6] = %s\nValuations: %s"
% ([0, s, t], [a1, a2, a3, a4, a6],
[pval(a1), pval(a2),
pval(a3), pval(a4),
pval(a6)]), t, 2)
if pval(a1) == 0:
raise RuntimeError, "p does not divide a1 after second transform!"
if pval(a2) == 0:
raise RuntimeError, "p does not divide a2 after second transform!"
if pval(a3) < 2:
raise RuntimeError, "p^2 does not divide a3 after second transform!"
if pval(a4) < 2:
raise RuntimeError, "p^2 does not divide a4 after second transform!"
if pval(a6) < 3:
raise RuntimeError, "p^3 does not divide a6 after second transform!"
if min(pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)) < 0:
raise RuntimeError, "Non-integral model after second transform!"
# Analyze roots of the cubic T^3 + bT^2 + cT + d = 0 mod P, where
# b = a2/p, c = a4/p^2, d = a6/p^3
b = preduce(a2 / pi)
c = preduce(a4 / pi2)
d = preduce(a6 / pi3)
bb = b * b
cc = c * c
bc = b * c
w = 27 * d * d - bb * cc + 4 * b * bb * d - 18 * bc * d + 4 * c * cc
x = 3 * c - bb
if pdiv(w):
if pdiv(x):
sw = 3
else:
sw = 2
else:
sw = 1
verbose(
"Analyzing roots of cubic T^3 + %s*T^2 + %s*T + %s, case %s" %
(b, c, d, sw), t, 1)
if sw == 1:
## Three distinct roots - Type I*0
verbose("Distinct roots", t, 1)
KS = KodairaSymbol("I0*")
cp = 1 + _pcubicroots(b, c, d)
fp = val_disc - 4
break #return
elif sw == 2:
## One double root - Type I*m for some m
verbose("One double root", t, 1)
## Change coords so that the double root is T = 0 mod p
if p == 2:
r = proot(c, 2)
elif p == 3:
r = c * pinv(b)
else:
r = (bc - 9 * d) * pinv(2 * x)
r = pi * preduce(r)
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
# The rest of this branch is just to compute cp, fp, KS.
# We use pi to keep transforms integral.
ix = 3
iy = 3
mx = pi2
my = mx
while True:
a2t = preduce(a2 / pi)
a3t = preduce(a3 / my)
a4t = preduce(a4 / (pi * mx))
a6t = preduce(a6 / (mx * my))
if pdiv(a3t * a3t + 4 * a6t):
if p == 2:
t = my * proot(a6t, 2)
else:
t = my * preduce(-a3t * halfmodp)
C = C.rst_transform(0, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
my *= pi
iy += 1
a2t = preduce(a2 / pi)
a3t = preduce(a3 / my)
a4t = preduce(a4 / (pi * mx))
a6t = preduce(a6 / (mx * my))
if pdiv(a4t * a4t - 4 * a6t * a2t):
if p == 2:
r = mx * proot(a6t * pinv(a2t), 2)
else:
r = mx * preduce(-a4t * pinv(2 * a2t))
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
mx *= pi
ix += 1 # and stay in loop
else:
if _pquadroots(a2t, a4t, a6t):
cp = 4
else:
cp = 2
break # exit loop
else:
if _pquadroots(1, a3t, -a6t):
cp = 4
else:
cp = 2
break
KS = KodairaSymbol("I%s*" % (ix + iy - 5))
fp = val_disc - ix - iy + 1
break #return
else: # sw == 3
## The cubic has a triple root
verbose("Triple root", t, 1)
## First we change coordinates so that T = 0 mod p
if p == 2:
r = b
elif p == 3:
r = proot(-d, 3)
else:
r = -b * pinv(3)
r = pi * preduce(r)
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
verbose(
"After third transform %s\n[a1,a2,a3,a4,a6] = %s\nValuations: %s"
% ([r, 0, 0], [a1, a2, a3, a4, a6],
[pval(ai) for ai in [a1, a2, a3, a4, a6]]), t, 2)
if min(pval(ai) for ai in [a1, a2, a3, a4, a6]) < 0:
raise RuntimeError, "Non-integral model after third transform!"
if pval(a2) < 2 or pval(a4) < 3 or pval(a6) < 4:
raise RuntimeError, "Cubic after transform does not have a triple root at 0"
a3t = preduce(a3 / pi2)
a6t = preduce(a6 / pi4)
# We test for Type IV*
if not pdiv(a3t * a3t + 4 * a6t):
cp = 3 if _pquadroots(1, a3t, -a6t) else 1
KS = KodairaSymbol("IV*")
fp = val_disc - 6
break #return
# Now change coordinates so that p^3|a3, p^5|a6
if p == 2:
t = -pi2 * proot(a6t, 2)
else:
t = pi2 * preduce(-a3t * halfmodp)
C = C.rst_transform(0, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
# We test for types III* and II*
if pval(a4) < 4:
## Type III*
KS = KodairaSymbol("III*")
fp = val_disc - 7
cp = 2
break #return
if pval(a6) < 6:
## Type II*
KS = KodairaSymbol("II*")
fp = val_disc - 8
cp = 1
break #return
if pi_neg is None:
if principal_flag:
pi_neg = pi
else:
pi_neg = K.uniformizer(P, 'negative')
pi_neg2 = pi_neg * pi_neg
pi_neg3 = pi_neg * pi_neg2
pi_neg4 = pi_neg * pi_neg3
pi_neg6 = pi_neg4 * pi_neg2
a1 /= pi_neg
a2 /= pi_neg2
a3 /= pi_neg3
a4 /= pi_neg4
a6 /= pi_neg6
verbose(
"Non-minimal equation, dividing out...\nNew model is %s" %
([a1, a2, a3, a4, a6]), t, 1)
return (C, p, val_disc, fp, KS, cp, split)
def descend_to(self, K, f=None):
r"""
Given a subfield `K` and an elliptic curve self defined over a field `L`,
this function determines whether there exists an elliptic curve over `K`
which is isomorphic over `L` to self. If one exists, it finds it.
INPUT:
- `K` -- a subfield of the base field of self.
- `f` -- an embedding of `K` into the base field of self.
OUTPUT:
Either an elliptic curve defined over `K` which is isomorphic to self
or None if no such curve exists.
.. NOTE::
This only works over number fields and QQ.
EXAMPLES::
sage: E = EllipticCurve([1,2,3,4,5])
sage: E.descend_to(ZZ)
Traceback (most recent call last):
...
TypeError: Input must be a field.
::
sage: F.<b> = QuadraticField(23)
sage: G.<a> = F.extension(x^3+5)
sage: E = EllipticCurve(j=1728*b).change_ring(G)
sage: E.descend_to(F)
Elliptic Curve defined by y^2 = x^3 + (8957952*b-206032896)*x + (-247669456896*b+474699792384) over Number Field in b with defining polynomial x^2 - 23
::
sage: L.<a> = NumberField(x^4 - 7)
sage: K.<b> = NumberField(x^2 - 7)
sage: E = EllipticCurve([a^6,0])
sage: E.descend_to(K)
Elliptic Curve defined by y^2 = x^3 + 1296/49*b*x over Number Field in b with defining polynomial x^2 - 7
::
sage: K.<a> = QuadraticField(17)
sage: E = EllipticCurve(j = 2*a)
sage: print E.descend_to(QQ)
None
"""
if not K.is_field():
raise TypeError, "Input must be a field."
if self.base_field() == K:
return self
j = self.j_invariant()
from sage.rings.all import QQ
if K == QQ:
f = QQ.embeddings(self.base_field())[0]
if j in QQ:
jbase = QQ(j)
else:
return None
elif f == None:
embeddings = K.embeddings(self.base_field())
if len(embeddings) == 0:
raise TypeError, "Input must be a subfield of the base field of the curve."
for g in embeddings:
try:
jbase = g.preimage(j)
f = g
break
except StandardError:
pass
if f == None:
return None
else:
try:
jbase = f.preimage(j)
except StandardError:
return None
E = EllipticCurve(j=jbase)
E2 = EllipticCurve(self.base_field(), [f(a) for a in E.a_invariants()])
if jbase == 0:
d = self.is_sextic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.sextic_twist(d)
elif jbase == 1728:
d = self.is_quartic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quartic_twist(d)
else:
d = self.is_quadratic_twist(E2)
if d == 1:
return E
if d == 0:
return None
Etwist = E2.quadratic_twist(d)
if Etwist.is_isomorphic(self):
try:
Eout = EllipticCurve(
K, [f.preimage(a) for a in Etwist.a_invariants()])
except StandardError:
return None
else:
return Eout