Python constructor.EllipticCurve类代码示例

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_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:
#.........这里部分代码省略.........

    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)
#.........这里部分代码省略.........

    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:
#.........这里部分代码省略.........

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 __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 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