def egros_from_j_0(S=[]):
r"""
Given a list of primes S, returns a list of elliptic curves over Q
with j-invariant 0 and good reduction outside S, by checking all
relevant sextic twists.
INPUT:
- S - list of primes (default: empty list).
.. note::
Primality of elements of S is not checked, and the output
is undefined if S is not a list or contains non-primes.
OUTPUT:
A sorted list of all elliptic curves defined over `Q` with
`j`-invariant equal to `0` and with good reduction at
all primes outside the list ``S``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_0
sage: egros_from_j_0([])
[]
sage: egros_from_j_0([2])
[]
sage: [e.label() for e in egros_from_j_0([3])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
sage: len(egros_from_j_0([2,3,5]))
432
"""
Elist = []
if not 3 in S:
return Elist
no2 = not 2 in S
for ei in xmrange([2] + [6] * len(S)):
u = prod([p**e for p, e in zip([-1] + S, ei)], QQ(1))
if no2:
u *= 16 ## make sure 12|val(D,2)
Eu = EllipticCurve([0, 0, 0, 0, u]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_0(S=[]):
r"""
Given a list of primes S, returns a list of elliptic curves over `\QQ`
with j-invariant 0 and good reduction outside S, by checking all
relevant sextic twists.
INPUT:
- S -- list of primes (default: empty list).
.. note::
Primality of elements of S is not checked, and the output
is undefined if S is not a list or contains non-primes.
OUTPUT:
A sorted list of all elliptic curves defined over `\QQ` with
`j`-invariant equal to `0` and with good reduction at
all primes outside the list ``S``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_0
sage: egros_from_j_0([])
[]
sage: egros_from_j_0([2])
[]
sage: [e.label() for e in egros_from_j_0([3])]
['27a1', '27a3', '243a1', '243a2', '243b1', '243b2']
sage: len(egros_from_j_0([2,3,5])) # long time (8s on sage.math, 2013)
432
"""
Elist=[]
if not 3 in S:
return Elist
no2 = not 2 in S
for ei in xmrange([2] + [6]*len(S)):
u = prod([p**e for p,e in zip([-1]+S,ei)],QQ(1))
if no2:
u*=16 ## make sure 12|val(D,2)
Eu = EllipticCurve([0,0,0,0,u]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_1728(S=[]):
r"""
Given a list of primes S, returns a list of elliptic curves over `\QQ`
with j-invariant 1728 and good reduction outside S, by checking
all relevant quartic twists.
INPUT:
- S -- list of primes (default: empty list).
.. note::
Primality of elements of S is not checked, and the output
is undefined if S is not a list or contains non-primes.
OUTPUT:
A sorted list of all elliptic curves defined over `\QQ` with
`j`-invariant equal to `1728` and with good reduction at
all primes outside the list ``S``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
sage: egros_from_j_1728([])
[]
sage: egros_from_j_1728([3])
[]
sage: [e.cremona_label() for e in egros_from_j_1728([2])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']
"""
Elist = []
no2 = not 2 in S
for ei in xmrange([2] + [4] * len(S)):
u = prod([p**e for p, e in zip([-1] + S, ei)], QQ(1))
if no2:
u *= 4 ## make sure 12|val(D,2)
Eu = EllipticCurve([0, 0, 0, u, 0]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def egros_from_j_1728(S=[]):
r"""
Given a list of primes S, returns a list of elliptic curves over `\QQ`
with j-invariant 1728 and good reduction outside S, by checking
all relevant quartic twists.
INPUT:
- S -- list of primes (default: empty list).
.. note::
Primality of elements of S is not checked, and the output
is undefined if S is not a list or contains non-primes.
OUTPUT:
A sorted list of all elliptic curves defined over `\QQ` with
`j`-invariant equal to `1728` and with good reduction at
all primes outside the list ``S``.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_from_j_1728
sage: egros_from_j_1728([])
[]
sage: egros_from_j_1728([3])
[]
sage: [e.cremona_label() for e in egros_from_j_1728([2])]
['32a1', '32a2', '64a1', '64a4', '256b1', '256b2', '256c1', '256c2']
"""
Elist=[]
no2 = not 2 in S
for ei in xmrange([2] + [4]*len(S)):
u = prod([p**e for p,e in zip([-1]+S,ei)],QQ(1))
if no2:
u*=4 ## make sure 12|val(D,2)
Eu = EllipticCurve([0,0,0,u,0]).minimal_model()
if Eu.has_good_reduction_outside_S(S):
Elist += [Eu]
Elist.sort(cmp=curve_cmp)
return Elist
def sextic_twist(self, D):
r"""
Return the quartic twist of this curve by `D`.
INPUT:
- ``D`` (must be nonzero) -- the twisting parameter..
.. note::
The characteristic must not be 2 or 3, and the `j`-invariant must be 0.
EXAMPLES::
sage: E=EllipticCurve_from_j(GF(13)(0)); E
Elliptic Curve defined by y^2 = x^3 + 1 over Finite Field of size 13
sage: E1=E.sextic_twist(2); E1
Elliptic Curve defined by y^2 = x^3 + 11 over Finite Field of size 13
sage: E.is_isomorphic(E1)
False
sage: E.is_isomorphic(E1,GF(13^2,'a'))
False
sage: E.is_isomorphic(E1,GF(13^4,'a'))
False
sage: E.is_isomorphic(E1,GF(13^6,'a'))
True
"""
K = self.base_ring()
char = K.characteristic()
D = K(D)
if char == 2 or char == 3:
raise ValueError("Sextic twist not defined in chars 2,3")
if self.j_invariant() != K(0):
raise ValueError("Sextic twist not defined when j!=0")
if D.is_zero():
raise ValueError("Sextic twist requires a nonzero argument")
c4, c6 = self.c_invariants()
# E is isomorphic to [0,0,0,0,-54*c6]
assert c4 == 0
return EllipticCurve(K, [0, 0, 0, 0, -54 * c6 * D])
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 descend_to(self, K, f=None):
r"""
Given an elliptic curve self defined over a field `L` and a
subfield `K` of `L`, return all elliptic curves over `K` which
are isomorphic over `L` to self.
INPUT:
- `K` -- a field which embeds into the base field `L` of self.
- `f` (optional) -- an embedding of `K` into `L`. Ignored if
`K` is `\QQ`.
OUTPUT:
A list (possibly empty) of elliptic curves defined over `K`
which are isomorphic to self over `L`, up to isomorphism over
`K`.
.. NOTE::
Currently only implemented over number fields. To extend
to other fields of characteristic not 2 or 3, what is
needed is a method giving the preimages in `K^*/(K^*)^m` of
an element of the base field, for `m=2,4,6`.
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: EF = E.descend_to(F); EF
[Elliptic Curve defined by y^2 = x^3 + (27*b-621)*x + (-1296*b+2484) over Number Field in b with defining polynomial x^2 - 23]
sage: all([Ei.change_ring(G).is_isomorphic(E) for Ei in EF])
True
::
sage: L.<a> = NumberField(x^4 - 7)
sage: K.<b> = NumberField(x^2 - 7, embedding=a^2)
sage: E = EllipticCurve([a^6,0])
sage: EK = E.descend_to(K); EK
[Elliptic Curve defined by y^2 = x^3 + b*x over Number Field in b with defining polynomial x^2 - 7,
Elliptic Curve defined by y^2 = x^3 + 7*b*x over Number Field in b with defining polynomial x^2 - 7]
sage: all([Ei.change_ring(L).is_isomorphic(E) for Ei in EK])
True
::
sage: K.<a> = QuadraticField(17)
sage: E = EllipticCurve(j = 2*a)
sage: E.descend_to(QQ)
[]
TESTS:
Check that :trac:`16456` is fixed::
sage: K.<a> = NumberField(x^3-2)
sage: E = EllipticCurve('11a1').quadratic_twist(2)
sage: EK = E.change_ring(K)
sage: EK2 = EK.change_weierstrass_model((a,a,a,a+1))
sage: EK2.descend_to(QQ)
[Elliptic Curve defined by y^2 = x^3 + x^2 - 41*x - 199 over Rational Field]
sage: k.<i> = QuadraticField(-1)
sage: E = EllipticCurve(k,[0,0,0,1,0])
sage: E.descend_to(QQ)
[Elliptic Curve defined by y^2 = x^3 + x over Rational Field,
Elliptic Curve defined by y^2 = x^3 - 4*x over Rational Field]
"""
if not K.is_field():
raise TypeError("Input must be a field.")
L = self.base_field()
if L is K:
return self
elif L == K: # number fields can be equal but not identical
return self.base_extend(K)
# Construct an embedding f of K in L, and check that the
# j-invariant is in the image, otherwise return an empty list:
j = self.j_invariant()
from sage.rings.all import QQ
if K == QQ:
try:
jK = QQ(j)
except (ValueError, TypeError):
return []
elif f is None:
embeddings = K.embeddings(L)
if len(embeddings) == 0:
raise TypeError(
"Input must be a subfield of the base field of the curve.")
for g in embeddings:
try:
jK = g.preimage(j)
f = g
break
except Exception:
pass
if f is None:
return []
else:
try:
if f.domain() != K:
raise ValueError("embedding has wrong domain")
if f.codomain() != L:
raise ValueError("embedding has wrong codomain")
except AttributeError:
raise ValueError("invalid embedding: %s" % s)
try:
jK = f.preimage(j)
except Exception:
return []
# Now we have the j-invariant in K and must find all twists
# which work, separating the cases of j=0 and j=1728.
if L.characteristic():
raise NotImplementedError(
"Not implemented in positive characteristic")
if jK == 0:
t = -54 * self.c6()
try:
dlist = t.descend_mod_power(K, 6)
# list of d in K such that t/d is in L*^6
except AttributeError:
raise NotImplementedError("Not implemented over %s" % L)
Elist = [EllipticCurve([0, 0, 0, 0, d]) for d in dlist]
elif jK == 1728:
t = -27 * self.c4()
try:
dlist = t.descend_mod_power(K, 4)
# list of d in K such that t/d is in L*^4
except AttributeError:
raise NotImplementedError("Not implemented over %s" % L)
Elist = [EllipticCurve([0, 0, 0, d, 0]) for d in dlist]
else:
c4, c6 = self.c_invariants()
t = c6 / c4
try:
dlist = t.descend_mod_power(K, 2)
# list of d in K such that t/d is in L*^2
except AttributeError:
raise NotImplementedError("Not implemented over %s" % L)
c = -27 * jK / (jK - 1728) # =-27c4^3/c6^2
a4list = [c * d**2 for d in dlist]
a6list = [2 * a4 * d for a4, d in zip(a4list, dlist)]
Elist = [
EllipticCurve([0, 0, 0, a4, a6])
for a4, a6 in zip(a4list, a6list)
]
if K is QQ:
Elist = [E.minimal_model() for E in Elist]
return Elist
def quadratic_twist(self, D=None):
"""
Return the quadratic twist of this curve by ``D``.
INPUT:
- ``D`` (default None) the twisting parameter (see below).
In characteristics other than 2, `D` must be nonzero, and the
twist is isomorphic to self after adjoining `\sqrt(D)` to the
base.
In characteristic 2, `D` is arbitrary, and the twist is
isomorphic to self after adjoining a root of `x^2+x+D` to the
base.
In characteristic 2 when `j=0`, this is not implemented.
If the base field `F` is finite, `D` need not be specified,
and the curve returned is the unique curve (up to isomorphism)
defined over `F` isomorphic to the original curve over the
quadratic extension of `F` but not over `F` itself. Over
infinite fields, an error is raised if `D` is not given.
EXAMPLES::
sage: E = EllipticCurve([GF(1103)(1), 0, 0, 107, 340]); E
Elliptic Curve defined by y^2 + x*y = x^3 + 107*x + 340 over Finite Field of size 1103
sage: F=E.quadratic_twist(-1); F
Elliptic Curve defined by y^2 = x^3 + 1102*x^2 + 609*x + 300 over Finite Field of size 1103
sage: E.is_isomorphic(F)
False
sage: E.is_isomorphic(F,GF(1103^2,'a'))
True
A characteristic 2 example::
sage: E=EllipticCurve(GF(2),[1,0,1,1,1])
sage: E1=E.quadratic_twist(1)
sage: E.is_isomorphic(E1)
False
sage: E.is_isomorphic(E1,GF(4,'a'))
True
Over finite fields, the twisting parameter may be omitted::
sage: k.<a> = GF(2^10)
sage: E = EllipticCurve(k,[a^2,a,1,a+1,1])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 + x*y = x^3 + (a^7+a^4+a^3+a^2+a+1)*x^2 + (a^8+a^6+a^4+1) over Finite Field in a of size 2^10
sage: E.is_isomorphic(Et)
False
sage: E.j_invariant()==Et.j_invariant()
True
sage: p=next_prime(10^10)
sage: k = GF(p)
sage: E = EllipticCurve(k,[1,2,3,4,5])
sage: Et = E.quadratic_twist()
sage: Et # random (only determined up to isomorphism)
Elliptic Curve defined by y^2 = x^3 + 7860088097*x^2 + 9495240877*x + 3048660957 over Finite Field of size 10000000019
sage: E.is_isomorphic(Et)
False
sage: k2 = GF(p^2,'a')
sage: E.change_ring(k2).is_isomorphic(Et.change_ring(k2))
True
"""
K = self.base_ring()
char = K.characteristic()
if D is None:
if K.is_finite():
x = rings.polygen(K)
if char == 2:
# We find D such that x^2+x+D is irreducible. If the
# degree is odd we can take D=1; otherwise it suffices to
# consider odd powers of a generator.
D = K(1)
if K.degree() % 2 == 0:
D = K.gen()
a = D**2
while len((x**2 + x + D).roots()) > 0:
D *= a
else:
# We could take a multiplicative generator but
# that might be expensive to compute; otherwise
# half the elements will do
D = K.random_element()
while len((x**2 - D).roots()) > 0:
D = K.random_element()
else:
raise ValueError(
"twisting parameter D must be specified over infinite fields."
)
else:
try:
D = K(D)
except ValueError:
raise ValueError(
"twisting parameter D must be in the base field.")
if char != 2 and D.is_zero():
raise ValueError(
"twisting parameter D must be nonzero when characteristic is not 2"
)
if char != 2:
b2, b4, b6, b8 = self.b_invariants()
# E is isomorphic to [0,b2,0,8*b4,16*b6]
return EllipticCurve(K,
[0, b2 * D, 0, 8 * b4 * D**2, 16 * b6 * D**3])
# now char==2
if self.j_invariant() != 0: # iff a1!=0
a1, a2, a3, a4, a6 = self.ainvs()
E0 = self.change_weierstrass_model(a1, a3 / a1, 0,
(a1**2 * a4 + a3**2) / a1**3)
# which has the form = [1,A2,0,0,A6]
assert E0.a1() == K(1)
assert E0.a3() == K(0)
assert E0.a4() == K(0)
return EllipticCurve(K, [1, E0.a2() + D, 0, 0, E0.a6()])
else:
raise ValueError(
"Quadratic twist not implemented in char 2 when j=0")
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 mod5family(a, b):
"""
Formulas for computing the family of elliptic curves with
congruent mod-5 representation.
AUTHORS:
-- Alice Silverberg and Karl Rubin (original PARI/GP version)
-- William Stein -- Sage version.
"""
J = 4 * a**3 / (4 * a**3 + 27 * b**2)
alpha = [0 for _ in range(21)]
alpha[0] = 1
alpha[1] = 0
alpha[2] = 190 * (J - 1)
alpha[3] = -2280 * (J - 1)**2
alpha[4] = 855 * (J - 1)**2 * (-17 + 16 * J)
alpha[5] = 3648 * (J - 1)**3 * (17 - 9 * J)
alpha[6] = 11400 * (J - 1)**3 * (17 - 8 * J)
alpha[7] = -27360 * (J - 1)**4 * (17 + 26 * J)
alpha[8] = 7410 * (J - 1)**4 * (-119 - 448 * J + 432 * J**2)
alpha[9] = 79040 * (J - 1)**5 * (17 + 145 * J - 108 * J**2)
alpha[10] = 8892 * (J - 1)**5 * (187 + 2640 * J - 5104 * J**2 +
1152 * J**3)
alpha[11] = 98800 * (J - 1)**6 * (-17 - 388 * J + 864 * J**2)
alpha[12] = 7410 * (J - 1)**6 * (-187 - 6160 * J + 24464 * J**2 -
24192 * J**3)
alpha[13] = 54720 * (J - 1)**7 * (17 + 795 * J - 3944 * J**2 + 9072 * J**3)
alpha[14] = 2280 * (J - 1)**7 * (221 + 13832 * J - 103792 * J**2 +
554112 * J**3 - 373248 * J**4)
alpha[15] = 1824 * (J - 1)**8 * (-119 - 9842 * J + 92608 * J**2 -
911520 * J**3 + 373248 * J**4)
alpha[16] = 4275 * (J - 1)**8 * (-17 - 1792 * J + 23264 * J**2 -
378368 * J**3 + 338688 * J**4)
alpha[17] = 18240 * (J - 1)**9 * (1 + 133 * J - 2132 * J**2 +
54000 * J**3 - 15552 * J**4)
alpha[18] = 190 * (J - 1)**9 * (17 + 2784 * J - 58080 * J**2 + 2116864 *
J**3 - 946944 * J**4 + 2985984 * J**5)
alpha[19] = 360 * (J - 1)**10 * (-1 + 28 * J - 1152 * J**2) * (
1 + 228 * J + 176 * J**2 + 1728 * J**3)
alpha[20] = (J - 1)**10 * (-19 - 4560 * J + 144096 * J**2 -
9859328 * J**3 - 8798976 * J**4 -
226934784 * J**5 + 429981696 * J**6)
beta = [0 for _ in range(31)]
beta[0] = 1
beta[1] = 30
beta[2] = -435 * (J - 1)
beta[3] = 580 * (J - 1) * (-7 + 9 * J)
beta[4] = 3915 * (J - 1)**2 * (7 - 8 * J)
beta[5] = 1566 * (J - 1)**2 * (91 - 78 * J + 48 * J**2)
beta[6] = -84825 * (J - 1)**3 * (7 + 16 * J)
beta[7] = 156600 * (J - 1)**3 * (-13 - 91 * J + 92 * J**2)
beta[8] = 450225 * (J - 1)**4 * (13 + 208 * J - 144 * J**2)
beta[9] = 100050 * (J - 1)**4 * (143 + 4004 * J - 5632 * J**2 +
1728 * J**3)
beta[10] = 30015 * (J - 1)**5 * (-1001 - 45760 * J + 44880 * J**2 -
6912 * J**3)
beta[11] = 600300 * (J - 1)**5 * (-91 - 6175 * J + 9272 * J**2 -
2736 * J**3)
beta[12] = 950475 * (J - 1)**6 * (91 + 8840 * J - 7824 * J**2)
beta[13] = 17108550 * (J - 1)**6 * (7 + 926 * J - 1072 * J**2 + 544 * J**3)
beta[14] = 145422675 * (J - 1)**7 * (-1 - 176 * J + 48 * J**2 - 384 * J**3)
beta[15] = 155117520 * (J - 1)**8 * (1 + 228 * J + 176 * J**2 +
1728 * J**3)
beta[16] = 145422675 * (J - 1)**8 * (1 + 288 * J + 288 * J**2 +
5120 * J**3 - 6912 * J**4)
beta[17] = 17108550 * (J - 1)**8 * (7 + 2504 * J + 3584 * J**2 + 93184 *
J**3 - 283392 * J**4 + 165888 * J**5)
beta[18] = 950475 * (J - 1)**9 * (-91 - 39936 * J - 122976 * J**2 -
2960384 * J**3 + 11577600 * J**4 -
5971968 * J**5)
beta[19] = 600300 * (J - 1)**9 * (-91 - 48243 * J - 191568 * J**2 -
6310304 * J**3 + 40515072 * J**4 -
46455552 * J**5 + 11943936 * J**6)
beta[20] = 30015 * (J - 1)**10 * (1001 + 634920 * J + 3880800 * J**2 +
142879744 * J**3 - 1168475904 * J**4 +
1188919296 * J**5 - 143327232 * J**6)
beta[21] = 100050 * (J - 1)**10 * (143 + 107250 * J + 808368 * J**2 +
38518336 * J**3 - 451953408 * J**4 +
757651968 * J**5 - 367276032 * J**6)
beta[22] = 450225 * (J - 1)**11 * (-13 - 11440 * J - 117216 * J**2 -
6444800 * J**3 + 94192384 * J**4 -
142000128 * J**5 + 95551488 * J**6)
beta[23] = 156600 * (J - 1)**11 * (
-13 - 13299 * J - 163284 * J**2 - 11171552 * J**3 + 217203840 * J**4 -
474406656 * J**5 + 747740160 * J**6 - 429981696 * J**7)
beta[24] = 6525*(J - 1)**12*(91 + 107536*J + 1680624*J**2 + 132912128*J**3 -\
3147511552*J**4 + 6260502528*J**5 - 21054173184*J**6 + 10319560704*J**7)
beta[25] = 1566*(J - 1)**12*(91 + 123292*J + 2261248*J**2 + 216211904*J**3 - \
6487793920*J**4 + 17369596928*J**5 - 97854234624*J**6 + 96136740864*J**7 - 20639121408*J**8)
beta[26] = 3915*(J - 1)**13*(-7 - 10816*J - 242352*J**2 - 26620160*J**3 + 953885440*J**4 - \
2350596096*J**5 + 26796552192*J**6 - 13329432576*J**7)
beta[27] = 580*(J - 1)**13*(-7 - 12259*J - 317176*J**2 - 41205008*J**3 + \
1808220160*J**4 - 5714806016*J**5 + 93590857728*J**6 - 70131806208*J**7 - 36118462464*J**8)
beta[28] = 435*(J - 1)**14*(1 + 1976*J + 60720*J**2 + 8987648*J**3 - 463120640*J**4 + 1359157248*J**5 - \
40644882432*J**6 - 5016453120*J**7 + 61917364224*J**8)
beta[29] = 30*(J - 1)**14*(1 + 2218*J + 77680*J**2 + 13365152*J**3 - \
822366976*J**4 + 2990693888*J**5 - 118286217216*J**6 - 24514928640*J**7 + 509958291456*J**8 - 743008370688*J**9)
beta[30] = (J - 1)**15*(-1 - 2480*J - 101040*J**2 - 19642496*J**3 + 1399023872*J**4 - \
4759216128*J**5 + 315623485440*J**6 + 471904911360*J**7 - 2600529297408*J**8 + 8916100448256*J**9)
R = PolynomialRing(QQ, 't')
b4 = a * R(alpha)
b6 = b * R(beta)
c2 = b4
c3 = b6
d = lcm(c2.denominator(), c3.denominator())
F = FractionField(R)
E = EllipticCurve(F, [c2 * d**4, c3 * d**6])
return E
def egros_get_j(S=[], proof=None, verbose=False):
r"""
Returns a list of rational `j` such that all elliptic curves
defined over `Q` with good reduction outside `S` have
`j`-invariant in the list, sorted by height.
INPUT:
- ``S`` - list of primes (default: empty list).
- ``proof`` - True/False (default True): the MW basis for
auxiliary curves will be computed with this proof flag.
- ``verbose`` - True/False (default False): if True, some
details of the computation will be output.
.. note::
Proof flag: The algorithm used requires determining all
S-integral points on several auxiliary curves, which in turn
requires the computation of their generators. This is not
always possible (even in theory) using current knowledge.
The value of this flag is passed to the function which
computes generators of various auxiliary elliptic curves, in
order to find their S-integral points. Set to False if the
default (True) causes warning messages, but note that you can
then not rely on the set of invariants returned being
complete.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j
sage: egros_get_j([])
[1728]
sage: egros_get_j([2])
[128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128]
sage: egros_get_j([3])
[0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000]
sage: jlist=egros_get_j([2,3]); len(jlist) # long time (30s)
83
"""
if not all([p.is_prime() for p in S]):
raise ValueError, "Elements of S must be prime."
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if verbose:
import sys # so we can flush stdout for debugging
SS = [-1] + S
jlist = []
wcount = 0
nw = 6**len(S) * 2
if verbose:
print "Finding possible j invariants for S = ", S
print "Using ", nw, " twists of base curve"
sys.stdout.flush()
for ei in xmrange([6] * len(S) + [2]):
w = prod([p**e for p, e in zip(reversed(SS), ei)], QQ(1))
wcount += 1
if verbose:
print "Curve #", wcount, "/", nw, ":"
print "w = ", w, "=", w.factor()
sys.stdout.flush()
a6 = -1728 * w
d2 = 0
d3 = 0
u0 = (2**d2) * (3**d3)
E = EllipticCurve([0, 0, 0, 0, a6])
# This curve may not be minimal at 2 or 3, but the
# S-integral_points function requires minimality at primes in
# S, so we find a new model which is p-minimal at both 2 and 3
# if they are in S. Note that the isomorphism between models
# will preserve S-integrality of points.
E2 = E.local_minimal_model(2) if 2 in S else E
E23 = E2.local_minimal_model(3) if 3 in S else E2
urst = E23.isomorphism_to(E)
try:
pts = E23.S_integral_points(S, proof=proof)
except RuntimeError:
pts = []
print "Failed to find S-integral points on ", E23.ainvs()
if proof:
if verbose:
print "--trying again with proof=False"
sys.stdout.flush()
pts = E23.S_integral_points(S, proof=False)
if verbose:
print "--done"
if verbose:
print len(pts), " S-integral points: ", pts
sys.stdout.flush()
for P in pts:
P = urst(P)
x = P[0]
y = P[1]
j = x**3 / w
assert j - 1728 == y**2 / w
if is_possible_j(j, S):
if not j in jlist:
if verbose:
print "Adding possible j = ", j
sys.stdout.flush()
jlist += [j]
else:
if True: #verbose:
print "Discarding illegal j = ", j
sys.stdout.flush()
height_cmp = lambda j1, j2: cmp(j1.height(), j2.height())
jlist.sort(cmp=height_cmp)
return jlist
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 rank(self, rank, tors=0, n=10, labels=False):
r"""
Return a list of at most `n` non-isogenous curves with given
rank and torsion order.
INPUT:
- ``rank`` (int) -- the desired rank
- ``tors`` (int, default 0) -- the desired torsion order (ignored if 0)
- ``n`` (int, default 10) -- the maximum number of curves returned.
- ``labels`` (bool, default False) -- if True, return Cremona
labels instead of curves.
OUTPUT:
(list) A list at most `n` of elliptic curves of required rank.
EXAMPLES::
sage: elliptic_curves.rank(n=5, rank=3, tors=2, labels=True)
['59450i1', '59450i2', '61376c1', '61376c2', '65481c1']
::
sage: elliptic_curves.rank(n=5, rank=0, tors=5, labels=True)
['11a1', '11a3', '38b1', '50b1', '50b2']
::
sage: elliptic_curves.rank(n=5, rank=1, tors=7, labels=True)
['574i1', '4730k1', '6378c1']
::
sage: e = elliptic_curves.rank(6)[0]; e.ainvs(), e.conductor()
((1, 1, 0, -2582, 48720), 5187563742)
sage: e = elliptic_curves.rank(7)[0]; e.ainvs(), e.conductor()
((0, 0, 0, -10012, 346900), 382623908456)
sage: e = elliptic_curves.rank(8)[0]; e.ainvs(), e.conductor()
((0, 0, 1, -23737, 960366), 457532830151317)
"""
from sage.misc.misc import SAGE_SHARE
db = os.path.join(SAGE_SHARE, 'ellcurves')
data = os.path.join(db, 'rank%s' % rank)
if not os.path.exists(data):
return []
v = []
tors = int(tors)
for w in open(data).readlines():
N, iso, num, ainvs, r, t = w.split()
if tors and tors != int(t):
continue
label = '%s%s%s' % (N, iso, num)
if labels:
v.append(label)
else:
E = EllipticCurve(eval(ainvs))
E._set_rank(r)
E._set_torsion_order(t)
E._set_conductor(N)
E._set_cremona_label(label)
v.append(E)
if len(v) >= n:
break
return v
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 _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 __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 simon_two_descent(E, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30):
"""
Interface to Simon's gp script for two-descent.
.. NOTE::
Users should instead run E.simon_two_descent()
EXAMPLES::
sage: import sage.schemes.elliptic_curves.gp_simon
sage: E=EllipticCurve('389a1')
sage: sage.schemes.elliptic_curves.gp_simon.simon_two_descent(E)
[2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)]]
sage: E.simon_two_descent()
(2, 2, [(1 : 0 : 1), (-11/9 : -55/27 : 1)])
TESTS::
sage: E = EllipticCurve('37a1').change_ring(QuadraticField(-11,'x'))
sage: E.simon_two_descent()
(1, 1, [(-1 : 0 : 1)])
"""
init()
current_randstate().set_seed_gp(gp)
K = E.base_ring()
K_orig = K
# The following is to correct the bug at \#5204: the gp script
# fails when K is a number field whose generator is called 'x'.
if not K is QQ:
K = K.change_names('a')
E_orig = E
E = EllipticCurve(K,[K(list(a)) for a in E.ainvs()])
F = E.integral_model()
if K != QQ:
# Simon's program requires that this name be y.
with localvars(K.polynomial().parent(), 'y'):
gp.eval("K = bnfinit(%s);" % K.polynomial())
if verbose >= 2:
print "K = bnfinit(%s);" % K.polynomial()
gp.eval("%s = Mod(y,K.pol);" % K.gen())
if verbose >= 2:
print "%s = Mod(y,K.pol);" % K.gen()
if K == QQ:
cmd = 'ellrank([%s,%s,%s,%s,%s]);' % F.ainvs()
else:
cmd = 'bnfellrank(K, [%s,%s,%s,%s,%s]);' % F.ainvs()
gp('DEBUGLEVEL_ell=%s; LIM1=%s; LIM3=%s; LIMTRIV=%s; MAXPROB=%s; LIMBIGPRIME=%s;'%(
verbose, lim1, lim3, limtriv, maxprob, limbigprime))
if verbose >= 2:
print cmd
s = gp.eval('ans=%s;'%cmd)
if s.find("***") != -1:
raise RuntimeError, "\n%s\nAn error occurred while running Simon's 2-descent program"%s
if verbose > 0:
print s
v = gp.eval('ans')
if v=='ans': # then the call to ellrank() or bnfellrank() failed
return 'fail'
if verbose >= 2:
print "v = ", v
# pari represents field elements as Mod(poly, defining-poly)
# so this function will return the respective elements of K
def _gp_mod(*args):
return args[0]
ans = sage_eval(v, {'Mod': _gp_mod, 'y': K.gen(0)})
inv_transform = F.isomorphism_to(E)
ans[2] = [inv_transform(F(P)) for P in ans[2]]
ans[2] = [E_orig([K_orig(list(c)) for c in list(P)]) for P in ans[2]]
return ans
def egros_get_j(S=[], proof=None, verbose=False):
r"""
Returns a list of rational `j` such that all elliptic curves
defined over `\QQ` with good reduction outside `S` have
`j`-invariant in the list, sorted by height.
INPUT:
- ``S`` -- list of primes (default: empty list).
- ``proof`` -- ``True``/``False`` (default ``True``): the MW basis for
auxiliary curves will be computed with this proof flag.
- ``verbose`` -- ``True``/``False`` (default ``False````): if ``True``, some
details of the computation will be output.
.. note::
Proof flag: The algorithm used requires determining all
S-integral points on several auxiliary curves, which in turn
requires the computation of their generators. This is not
always possible (even in theory) using current knowledge.
The value of this flag is passed to the function which
computes generators of various auxiliary elliptic curves, in
order to find their S-integral points. Set to ``False`` if the
default (``True``) causes warning messages, but note that you can
then not rely on the set of invariants returned being
complete.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j
sage: egros_get_j([])
[1728]
sage: egros_get_j([2]) # long time (3s on sage.math, 2013)
[128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128]
sage: egros_get_j([3]) # long time (3s on sage.math, 2013)
[0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000]
sage: jlist=egros_get_j([2,3]); len(jlist) # long time (30s)
83
"""
if not all([p.is_prime() for p in S]):
raise ValueError("Elements of S must be prime.")
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if verbose:
import sys # so we can flush stdout for debugging
SS = [-1] + S
jlist=[]
wcount=0
nw = 6**len(S) * 2
if verbose:
print "Finding possible j invariants for S = ",S
print "Using ", nw, " twists of base curve"
sys.stdout.flush()
for ei in xmrange([6]*len(S) + [2]):
w = prod([p**e for p,e in zip(reversed(SS),ei)],QQ(1))
wcount+=1
if verbose:
print "Curve #",wcount, "/",nw,":";
print "w = ",w,"=",w.factor()
sys.stdout.flush()
a6 = -1728*w
d2 = 0
d3 = 0
u0 = (2**d2)*(3**d3)
E = EllipticCurve([0,0,0,0,a6])
# This curve may not be minimal at 2 or 3, but the
# S-integral_points function requires minimality at primes in
# S, so we find a new model which is p-minimal at both 2 and 3
# if they are in S. Note that the isomorphism between models
# will preserve S-integrality of points.
E2 = E.local_minimal_model(2) if 2 in S else E
E23 = E2.local_minimal_model(3) if 3 in S else E2
urst = E23.isomorphism_to(E)
try:
pts = E23.S_integral_points(S,proof=proof)
except RuntimeError:
pts=[]
print "Failed to find S-integral points on ",E23.ainvs()
if proof:
if verbose:
print "--trying again with proof=False"
sys.stdout.flush()
pts = E23.S_integral_points(S,proof=False)
if verbose:
print "--done"
if verbose:
print len(pts), " S-integral points: ",pts
sys.stdout.flush()
for P in pts:
P = urst(P)
x = P[0]
y = P[1]
j = x**3 /w
assert j-1728 == y**2 /w
if is_possible_j(j,S):
if not j in jlist:
if verbose:
print "Adding possible j = ",j
sys.stdout.flush()
jlist += [j]
else:
if True: #verbose:
print "Discarding illegal j = ",j
sys.stdout.flush()
return sorted(jlist, key=lambda j: j.height())
def rank(self, rank, tors=0, n=10, labels=False):
r"""
Return a list of at most `n` non-isogenous curves with given
rank and torsion order.
INPUT:
- ``rank`` (int) -- the desired rank
- ``tors`` (int, default 0) -- the desired torsion order (ignored if 0)
- ``n`` (int, default 10) -- the maximum number of curves returned.
- ``labels`` (bool, default False) -- if True, return Cremona
labels instead of curves.
OUTPUT:
(list) A list at most `n` of elliptic curves of required rank.
EXAMPLES::
sage: elliptic_curves.rank(n=5, rank=3, tors=2, labels=True)
['59450i1', '59450i2', '61376c1', '61376c2', '65481c1']
::
sage: elliptic_curves.rank(n=5, rank=0, tors=5, labels=True)
['11a1', '11a3', '38b1', '50b1', '50b2']
::
sage: elliptic_curves.rank(n=5, rank=1, tors=7, labels=True)
['574i1', '4730k1', '6378c1']
::
sage: e = elliptic_curves.rank(6)[0]; e.ainvs(), e.conductor()
((1, 1, 0, -2582, 48720), 5187563742)
sage: e = elliptic_curves.rank(7)[0]; e.ainvs(), e.conductor()
((0, 0, 0, -10012, 346900), 382623908456)
sage: e = elliptic_curves.rank(8)[0]; e.ainvs(), e.conductor()
((0, 0, 1, -23737, 960366), 457532830151317)
"""
from sage.env import SAGE_SHARE
db = os.path.join(SAGE_SHARE, 'ellcurves')
data = os.path.join(db, 'rank%s' % rank)
if not os.path.exists(data):
return []
v = []
tors = int(tors)
for w in open(data).readlines():
N, iso, num, ainvs, r, t = w.split()
if tors and tors != int(t):
continue
label = '%s%s%s' % (N, iso, num)
if labels:
v.append(label)
else:
E = EllipticCurve(eval(ainvs))
E._set_rank(r)
E._set_torsion_order(t)
E._set_conductor(N)
E._set_cremona_label(label)
v.append(E)
if len(v) >= n:
break
return v
