本文整理匯總了Python中constructor.EllipticCurve類的典型用法代碼示例。如果您正苦於以下問題：Python EllipticCurve類的具體用法？Python EllipticCurve怎麽用？Python EllipticCurve使用的例子？那麽, 這裏精選的類代碼示例或許可以為您提供幫助。
在下文中一共展示了EllipticCurve類的8個代碼示例，這些例子默認根據受歡迎程度排序。您可以為喜歡或者感覺有用的代碼點讚，您的評價將有助於係統推薦出更棒的Python代碼示例。
示例1: egros_from_j_0
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示例2: egros_from_j_1728
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示例3: egros_get_j
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:
#.........這裏部分代碼省略.........
示例4: _tate
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)
#.........這裏部分代碼省略.........
示例5: descend_to
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:
#.........這裏部分代碼省略.........
示例6: simon_two_descent
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示例7: __init__
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示例8: rank
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