def mumu(N):
    """
    Return 0 if any cube divides `N`. Otherwise return
    `(-2)^v` where `v` is the number of primes that
    exactly divide `N`.

    This is similar to the Moebius function.

    INPUT:

    -  ``N`` - an integer at least 1

    OUTPUT: Integer

    EXAMPLES::

        sage: from sage.modular.arithgroup.congroup_gammaH import mumu
        sage: mumu(27)
        0
        sage: mumu(6*25)
        4
        sage: mumu(7*9*25)
        -2
        sage: mumu(9*25)
        1
    """
    if N < 1:
        raise ValueError, "N must be at least 1"
    p = 1
    for _,r in factor(N):
        if r > 2:
            return ZZ(0)
        elif r == 1:
            p *= -2
    return ZZ(p)

    def multiplicative_order(self):
        r"""
        Returns the *multiplicative* order of this element, which must be
        nonzero.

        EXAMPLES::

            sage: from sage.rings.finite_rings.finite_field_ext_pari import FiniteField_ext_pari
            sage: a = FiniteField_ext_pari(5**3, 'a').0
            sage: a.multiplicative_order()
            124
            sage: a**124
            1
        """
        try:
            return self.__multiplicative_order
        except AttributeError:
            if self.is_zero():
                raise ArithmeticError("Multiplicative order of 0 not defined.")
            n = self.parent().order() - 1
            order = 1
            for p, e in arith.factor(n):
                # Determine the power of p that divides the order.
                a = self**(n//(p**e))
                while a != 1:
                    order *= p
                    a = a**p
            self.__multiplicative_order = order
            return order

    def coefficient_n_recursive(self, n):
        r"""
          Reimplement the recursive algorithm in sage modular/hecke/module.py
          We do this because of a bug in sage with .eigenvalue()
        """
        from sage.rings import arith
        ev = self.eigenvalues

        c2 = self._coefficients.get(2)
        if c2 is not None:
            K = c2.parent()
        else:
            if ev.max_coefficient_in_db() >= 2:
                ev.init_dynamic_properties()
            else:
                raise StopIteration,"Newform does not have eigenvalue a(2)!"
            self._coefficients[2]=ev[2]
            K = ev[2].parent()
        prod = K(1)
        if K.absolute_degree()>1 and K.is_relative():
            KZ = K.base_field()
        else:
            KZ = K
        #emf_logger.debug("K= {0}".format(K))
        F = arith.factor(n)
        for p, r in F:
            #emf_logger.debug("parent_char_val[{0}]={1}".format(p,self.parent.character_used_in_computation.value(p)))
            #emf_logger.debug("char_val[{0}]={1}".format(p,self.character.value(p)))
            (p, r) = (int(p), int(r))
            pr = p**r
            cp = self._coefficients.get(p)
#            emf_logger.debug("c{0} = {1}".format(p,cp))
            if cp is None:
                if ev.has_eigenvalue(p):
                    cp = ev[p]
                elif ev.max_coefficient_in_db() >= p:
                    ev.init_dynamic_properties()
                    cp = ev[p]
            if cp is None:
                raise IndexError,"p={0} is outside the range of computed primes (primes up to {1})! for label:{2}".format(p,max(ev.primes()),self.label)
            if self._coefficients.get(pr) is None:
                if r == 1:
                    c = cp
                else:
                    eps = KZ(self.parent.character_used_in_computation.value(p))
                    # a_{p^r} := a_p * a_{p^{r-1}} - eps(p)p^{k-1} a_{p^{r-2}}
                    apr1 = self.coefficient_n_recursive(pr//p)
                    #ap = self.coefficient_n_recursive(p)
                    k = self.weight
                    apr2 = self.coefficient_n_recursive(pr//(p*p))
                    c = cp*apr1 - eps*(p**(k-1)) * apr2
                    #emf_logger.debug("c({0})={1}".format(pr,c))
                            #ev[pr]=c
                self._coefficients[pr]=c
            prod *= self._coefficients[pr]
        return prod

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None, impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), 'conway', None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), 'conway', None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic

        if proof is None:
            proof = arithmetic()
        with WithProof("arithmetic", proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None:
                    name = names
                name = normalize_names(1, name)

                p, n = arith.factor(order)[0]

                if modulus is None or modulus == "default":
                    if exists_conway_polynomial(p, n):
                        modulus = "conway"
                    else:
                        if p == 2:
                            modulus = "minimal_weight"
                        else:
                            modulus = "random"
                elif modulus == "random":
                    modulus += str(random.randint(0, 1 << 128))

                if isinstance(modulus, (list, tuple)):
                    modulus = FiniteField(p)["x"](modulus)
                # some classes use 'random' as the modulus to
                # generate a random modulus, but we don't want
                # to cache it
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name("x")
                elif not isinstance(modulus, str):
                    raise ValueError("Modulus parameter not understood.")
            else:  # Neither a prime, nor a prime power
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

    def factored_order(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: FF = IntegerModRing(17)
            sage: R.factored_order()
            2 * 3^2
            sage: FF.factored_order()
            17
        """
        return factor(self.__order, int_=(self.__order < 2 ** 31))

def irred(K, n):
    F = []
    p = K.characteristic()
    for ell, exp in factor(n):
        if ell == 2:
            F.append(rand_irred(K, ell**exp))
        elif ell.divides(p-1):
            F.append(kummer(K, ell, exp))
        elif ell.divides(p+1):
            F.append(cheby(K, ell, exp))
        else:
            F.append(rand_irred(K, ell**exp))

    return reduce(comp_prod, F)

    def factored_order(self):
        """
        EXAMPLES::

            sage: R = IntegerModRing(18)
            sage: FF = IntegerModRing(17)
            sage: R.factored_order()
            2 * 3^2
            sage: FF.factored_order()
            17
        """
        if self.__factored_order is not None:
            return self.__factored_order
        self.__factored_order = factor(self.__order, int_=(self.__order < 2**31))
        return self.__factored_order

def squarefree_part(x):
    """
    Returns the square free part of `x`, i.e., a divisor
    `z` such that `x = z y^2`, for a perfect square
    `y^2`.

    EXAMPLES::

        sage: squarefree_part(100)
        1
        sage: squarefree_part(12)
        3
        sage: squarefree_part(10)
        10
        sage: squarefree_part(216r) # see #8976
        6

    ::

        sage: x = QQ['x'].0
        sage: S = squarefree_part(-9*x*(x-6)^7*(x-3)^2); S
        -9*x^2 + 54*x
        sage: S.factor()
        (-9) * (x - 6) * x

    ::

        sage: f = (x^3 + x + 1)^3*(x-1); f
        x^10 - x^9 + 3*x^8 + 3*x^5 - 2*x^4 - x^3 - 2*x - 1
        sage: g = squarefree_part(f); g
        x^4 - x^3 + x^2 - 1
        sage: g.factor()
        (x - 1) * (x^3 + x + 1)
    """
    try:
        return x.squarefree_part()
    except AttributeError:
        pass
    from sage.rings.arith import factor
    from sage.structure.all import parent
    F = factor(x)
    n = parent(x)(1)
    for p, e in F:
        if e%2 != 0:
            n *= p
    return n * F.unit()

    def factored_unit_order(self):
        """
        Returns a list of Factorization objects, each the factorization of the
        order of the units in a `\ZZ / p^n \ZZ` component of this group (using
        the Chinese Remainder Theorem).

        EXAMPLES::

            sage: R = Integers(8*9*25*17*29)
            sage: R.factored_unit_order()
            [2^2, 2 * 3, 2^2 * 5, 2^4, 2^2 * 7]
        """
        ans = []
        from sage.structure.factorization import Factorization
        for p, e in self.factored_order():
            ans.append(Factorization([(p,e-1)]) * factor(p-1, int_=(self.__order < 2**31)))
        return ans

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
                                  impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic
        if proof is None: proof = arithmetic()
        with WithProof('arithmetic', proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None: name = names
                name = normalize_names(1,name)

                p, n = arith.factor(order)[0]

                if modulus is None or isinstance(modulus, str):
                    # A string specifies an algorithm to find a suitable modulus.
                    if modulus == "default":    # for backward compatibility
                        modulus = None
                    modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus)
                elif isinstance(modulus, (list, tuple)):
                    modulus = GF(p)['x'](modulus)
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name('x')
                else:
                    raise TypeError("wrong type for modulus parameter")
            else:
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

    def other_keys(self, key, K):
        """
        EXAMPLES::

            sage: key, extra = GF.create_key_and_extra_args(9, 'a'); key
            (9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True)
            sage: K = GF.create_object(0, key); K
            Finite Field in a of size 3^2
            sage: GF.other_keys(key, K)
            [(9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True),
             (9, ('a',), x^2 + 2*x + 2, 'givaro', '{}', 3, 2, True)]
        """
        if len(key) == 5: # backward compat
            order, name, modulus, impl, _ = key
            p, n = arith.factor(order)[0]
            proof = True
        else:
            order, name, modulus, impl, _, p, n, proof = key

        from sage.structure.proof.all import WithProof
        with WithProof('arithmetic', proof):
            if K.degree() > 1:
                modulus = K.modulus().change_variable_name('x')
            new_keys = [(order, name, modulus, impl, _, p, n, proof)]
            from finite_field_prime_modn import FiniteField_prime_modn
            if isinstance(K, FiniteField_prime_modn):
                impl = 'modn'
            elif isinstance(K, FiniteField_givaro):
                impl = 'givaro'
            else:
                from finite_field_ntl_gf2e import FiniteField_ntl_gf2e
                from finite_field_ext_pari import FiniteField_ext_pari
                from finite_field_pari_ffelt import FiniteField_pari_ffelt
                if isinstance(K, FiniteField_ntl_gf2e):
                    impl = 'ntl'
                elif isinstance(K, FiniteField_ext_pari):
                    impl = 'pari_mod'
                elif isinstance(K, FiniteField_pari_ffelt):
                    impl = 'pari_ffelt'
            new_keys.append( (order, name, modulus, impl, _, p, n, proof) )
            return new_keys

def clifford_conductor(self):
    """
    This is the product of all primes where the Clifford invariant is -1

    Note: For ternary forms, this is the discriminant of the
    quaternion algebra associated to the quadratic space
    (i.e. the even Clifford algebra)

    EXAMPLES::

        sage: Q = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
        sage: Q.clifford_invariant(2)
        1
        sage: Q.clifford_invariant(37)
        -1
        sage: Q.clifford_conductor()
        37

    ::

        sage: DiagonalQuadraticForm(ZZ, [1, 1, 1]).clifford_conductor()
        2
        sage: QuadraticForm(ZZ, 3, [2, -2, 0, 2, 0, 5]).clifford_conductor()
        30

    For hyperbolic spaces, the clifford conductor is 1::

        sage: H = QuadraticForm(ZZ, 2, [0, 1, 0])
        sage: H.clifford_conductor()
        1
        sage: (H + H).clifford_conductor()
        1
        sage: (H + H + H).clifford_conductor()
        1
        sage: (H + H + H + H).clifford_conductor()
        1

    """
    D = self.disc()
    return prod(filter(lambda p: self.clifford_invariant(p) == -1,
                       map(lambda x: x[0], factor(2 * self.level()))))

def hasse_conductor(self):
    """
    This is the product of all primes where the Hasse invariant equals -1

    EXAMPLES::

        sage: Q = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
        sage: Q.hasse_invariant(2)
        -1
        sage: Q.hasse_invariant(37)
        -1
        sage: Q.hasse_conductor()
        74

    ::

        sage: DiagonalQuadraticForm(ZZ, [1, 1, 1]).hasse_conductor()
        1
        sage: QuadraticForm(ZZ, 3, [2, -2, 0, 2, 0, 5]).hasse_conductor()
        10
    """
    D = self.disc()
    return prod([x[0] for x in factor(2 * self.level()) if self.hasse_invariant(x[0]) == -1])

def hasse_conductor(self):
    """
    This is the product of all primes where the Hasse invariant equals -1

    EXAMPLES::

        sage: Q = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
        sage: Q.hasse_invariant(2)
        -1
        sage: Q.hasse_invariant(37)
        -1
        sage: Q.hasse_conductor()
        74

    ::

        sage: DiagonalQuadraticForm(ZZ, [1, 1, 1]).hasse_conductor()
        1
        sage: QuadraticForm(ZZ, 3, [2, -2, 0, 2, 0, 5]).hasse_conductor()
        10
    """
    D = self.disc()
    return prod(filter(lambda(p):self.hasse_invariant(p)==-1, \
             map(lambda(x):x[0],factor(2*self.level()))))

def CohenOesterle(eps, k):
    r"""
    Compute the Cohen-Oesterle function associate to eps, `k`.
    This is a summand in the formula for the dimension of the space of
    cusp forms of weight `2` with character
    `\varepsilon`.

    INPUT:


    -  ``eps`` - Dirichlet character

    -  ``k`` - integer


    OUTPUT: element of the base ring of eps.

    EXAMPLES::

        sage: G.<eps> = DirichletGroup(7)
        sage: sage.modular.dims.CohenOesterle(eps, 2)
        -2/3
        sage: sage.modular.dims.CohenOesterle(eps, 4)
        -1
    """
    N    = eps.modulus()
    facN = factor(N)
    f    = eps.conductor()
    gamma_k = 0
    if k%4==2:
        gamma_k = frac(-1,4)
    elif k%4==0:
        gamma_k = frac(1,4)
    mu_k = 0
    if k%3==2:
        mu_k = frac(-1,3)
    elif k%3==0:
        mu_k = frac(1,3)
    def _lambda(r,s,p):
        """
        Used internally by the CohenOesterle function.

        INPUT:


        -  ``r, s, p`` - integers


        OUTPUT: Integer

        EXAMPLES: (indirect doctest)

        ::

            sage: K = CyclotomicField(3)
            sage: eps = DirichletGroup(7*43,K).0^2
            sage: sage.modular.dims.CohenOesterle(eps,2)
            -4/3
        """
        if 2*s<=r:
            if r%2==0:
                return p**(r//2) + p**((r//2)-1)
            return 2*p**((r-1)//2)
        return 2*(p**(r-s))
    #end def of lambda
    K = eps.base_ring()
    return K(frac(-1,2) * mul([_lambda(r,valuation(f,p),p) for p, r in facN]) + \
               gamma_k * mul([CO_delta(r,p,N,eps)         for p, r in facN]) + \
                mu_k    * mul([CO_nu(r,p,N,eps)            for p, r in facN]))

def O(*x):
    """
    Big O constructor for various types.

    EXAMPLES:

    This is useful for writing power series elements. ::

        sage: R.<t> = ZZ[['t']]
        sage: (1+t)^10 + O(t^5)
        1 + 10*t + 45*t^2 + 120*t^3 + 210*t^4 + O(t^5)

    A power series ring is created implicitly if a polynomial element is passed in. ::

        sage: R.<x> = QQ['x']
        sage: O(x^100)
        O(x^100)
        sage: 1/(1+x+O(x^5))
        1 - x + x^2 - x^3 + x^4 + O(x^5)
        sage: R.<u,v> = QQ[[]]
        sage: 1 + u + v^2 + O(u, v)^5
        1 + u + v^2 + O(u, v)^5

    This is also useful to create p-adic numbers. ::

        sage: O(7^6)
        O(7^6)
        sage: 1/3 + O(7^6)
        5 + 4*7 + 4*7^2 + 4*7^3 + 4*7^4 + 4*7^5 + O(7^6)

    It behaves well with respect to adding negative powers of p::

        sage: a = O(11^-32); a
        O(11^-32)
        sage: a.parent()
        11-adic Field with capped relative precision 20

    There are problems if you add a rational with very negative valuation to a big_oh. ::

        sage: 11^-12 + O(11^15)
        11^-12 + O(11^8)

    The reason that this fails is that the O function doesn't know the right precision cap to use.  If you cast explicitly or use other means of element creation, you can get around this issue. ::

        sage: K = Qp(11, 30)
        sage: K(11^-12) + O(11^15)
        11^-12 + O(11^15)
        sage: 11^-12 + K(O(11^15))
        11^-12 + O(11^15)
        sage: K(11^-12, absprec = 15)
        11^-12 + O(11^15)
        sage: K(11^-12, 15)
        11^-12 + O(11^15)

    """
    if len(x) > 1:
        if isinstance(x[0], multi_power_series_ring_element.MPowerSeries):
            return multi_power_series_ring_element.MO(x)
    x = x[0]

    if isinstance(x, power_series_ring_element.PowerSeries):
        return x.parent()(0, x.degree())

    elif isinstance(x, Polynomial):
        if x.parent().ngens() != 1:
            raise NotImplementedError("completion only currently defined for univariate polynomials")
        if not x.is_monomial():
            raise NotImplementedError("completion only currently defined for the maximal ideal (x)")
        return x.parent().completion(x.parent().gen())(0, x.degree())

    elif isinstance(x, laurent_series_ring_element.LaurentSeries):
        return laurent_series_ring_element.LaurentSeries(x.parent(), 0).add_bigoh(x.valuation())

    elif isinstance(x, (int,long,integer.Integer,rational.Rational)):  # p-adic number
        if x <= 0:
            raise ArithmeticError("x must be a prime power >= 2")
        F = arith.factor(x)
        if len(F) != 1:
            raise ArithmeticError("x must be prime power")
        p, r = F[0]
        if r >= 0:
            return padics_factory.Zp(p, prec = max(r, 20), type = 'capped-rel')(0, absprec = r)
        else:
            return padics_factory.Qp(p, prec = max(r, 20), type = 'capped-rel')(0, absprec = r)

    elif isinstance(x, padic_generic_element.pAdicGenericElement):
         return x.parent()(0, absprec = x.valuation())
    raise ArithmeticError("O(x) not defined")

def mutually_orthogonal_latin_squares(n,k=None, partitions = False):
    r"""
    Returns `k` Mutually Orthogonal `n\times n` Latin Squares (MOLS).

    For more information on Latin Squares and MOLS, see
    :mod:`~sage.combinat.designs.latin_squares` or the :wikipedia:`Latin_square`,
    or even the
    :wikipedia:`Wikipedia entry on MOLS <Graeco-Latin_square#Mutually_orthogonal_Latin_squares>`.

    INPUT:

    - ``n`` (integer) -- size of the latin square.

    - ``k`` (integer) -- returns `k` MOLS. If set to ``None`` (default), returns
      the maximum number of MOLS that Sage can build.

      .. WARNING::

          This has no reason to be the maximum number of `n\times n` MOLS, just
          the best Sage can do !

    - ``partition`` (boolean) -- a Latin Square can be seen as 3 partitions of
      the `n^2` cells of the array into `n` sets of size `n`, respectively :

      * The partition of rows
      * The partition of columns
      * The partition of number (cells numbered with 0, cells numbered with 1,
        ...)

      These partitions have the additional property that any two sets from
      different partitions intersect on exactly one element.

      When ``partition`` is set to ``True``, this function returns a list of `k+2`
      partitions satisfying this intersection property instead of the `k+2` MOLS
      (though the data is exactly the same in both cases).

    EXAMPLES::

        sage: designs.mutually_orthogonal_latin_squares(5)
        [
        [0 1 2 3 4]  [0 1 2 3 4]  [0 1 2 3 4]  [0 1 2 3 4]
        [3 0 1 4 2]  [4 3 0 2 1]  [1 2 4 0 3]  [2 4 3 1 0]
        [4 3 0 2 1]  [1 2 4 0 3]  [2 4 3 1 0]  [3 0 1 4 2]
        [1 2 4 0 3]  [2 4 3 1 0]  [3 0 1 4 2]  [4 3 0 2 1]
        [2 4 3 1 0], [3 0 1 4 2], [4 3 0 2 1], [1 2 4 0 3]
        ]
        sage: designs.mutually_orthogonal_latin_squares(7,3)
        [
        [0 1 2 3 4 5 6]  [0 1 2 3 4 5 6]  [0 1 2 3 4 5 6]
        [4 0 3 1 6 2 5]  [5 6 0 4 2 1 3]  [6 4 1 0 5 3 2]
        [5 6 0 4 2 1 3]  [6 4 1 0 5 3 2]  [1 3 5 2 0 6 4]
        [6 4 1 0 5 3 2]  [1 3 5 2 0 6 4]  [2 5 4 6 3 0 1]
        [1 3 5 2 0 6 4]  [2 5 4 6 3 0 1]  [3 2 6 5 1 4 0]
        [2 5 4 6 3 0 1]  [3 2 6 5 1 4 0]  [4 0 3 1 6 2 5]
        [3 2 6 5 1 4 0], [4 0 3 1 6 2 5], [5 6 0 4 2 1 3]
        ]
        sage: designs.mutually_orthogonal_latin_squares(5,2,partitions=True)
        [[[0, 1, 2, 3, 4],
          [5, 6, 7, 8, 9],
          [10, 11, 12, 13, 14],
          [15, 16, 17, 18, 19],
          [20, 21, 22, 23, 24]],
         [[0, 5, 10, 15, 20],
          [1, 6, 11, 16, 21],
          [2, 7, 12, 17, 22],
          [3, 8, 13, 18, 23],
          [4, 9, 14, 19, 24]],
        [[0, 6, 12, 18, 24],
          [1, 7, 14, 15, 23],
          [2, 9, 13, 16, 20],
          [3, 5, 11, 19, 22],
          [4, 8, 10, 17, 21]],
        [[0, 7, 13, 19, 21],
          [1, 9, 10, 18, 22],
          [2, 8, 11, 15, 24],
          [3, 6, 14, 17, 20],
          [4, 5, 12, 16, 23]]]

    TESTS::

        sage: designs.mutually_orthogonal_latin_squares(5,5)
        Traceback (most recent call last):
        ...
        ValueError: There exist at most n-1 MOLS of size n.
    """
    from sage.rings.finite_rings.constructor import FiniteField
    from sage.combinat.designs.block_design import AffineGeometryDesign
    from sage.rings.arith import is_prime_power
    from sage.matrix.constructor import Matrix
    from sage.rings.arith import factor

    if k is not None and k >= n:
        raise ValueError("There exist at most n-1 MOLS of size n.")

    if is_prime_power(n):
        if k is None:
            k = n-1
        # Section 6.4.1 of [Stinson2004]
        Fp = FiniteField(n,'x')
        B = AffineGeometryDesign(2,1,Fp).blocks()
#.........这里部分代码省略.........

    def create_object(self, version, key, check_irreducible=True, elem_cache=None,
                      names=None, **kwds):
        """
        EXAMPLES::

            sage: K = GF(19) # indirect doctest
            sage: TestSuite(K).run()
        """
        # IMPORTANT!  If you add a new class to the list of classes
        # that get cached by this factor object, then you *must* add
        # the following method to that class in order to fully support
        # pickling:
        #
        #     def __reduce__(self):   # and include good doctests, please!
        #         return self._factory_data[0].reduce_data(self)
        #
        # This is not in the base class for finite fields, since some finite
        # fields need not be created using this factory object, e.g., residue
        # class fields.

        if len(key) == 5:
            # for backward compatibility of pickles (see trac 10975).
            order, name, modulus, impl, _ = key
            p, n = arith.factor(order)[0]
            proof = True
        else:
            order, name, modulus, impl, _, p, n, proof = key

        if elem_cache is None:
            elem_cache = order < 500

        if n == 1 and (impl is None or impl == 'modn'):
            from finite_field_prime_modn import FiniteField_prime_modn
            # Using a check option here is probably a worthwhile
            # compromise since this constructor is simple and used a
            # huge amount.
            K = FiniteField_prime_modn(order, check=False)
        else:
            # We have to do this with block so that the finite field
            # constructors below will use the proof flag that was
            # passed in when checking for primality, factoring, etc.
            # Otherwise, we would have to complicate all of their
            # constructors with check options (like above).
            from sage.structure.proof.all import WithProof
            with WithProof('arithmetic', proof):
                if check_irreducible and polynomial_element.is_Polynomial(modulus):
                    if modulus.parent().base_ring().characteristic() == 0:
                        modulus = modulus.change_ring(FiniteField(p))
                    if not modulus.is_irreducible():
                        raise ValueError("finite field modulus must be irreducible but it is not.")
                    if modulus.degree() != n:
                        raise ValueError("The degree of the modulus does not correspond to the cardinality of the field.")
                if name is None:
                    raise TypeError("you must specify the generator name.")
                if impl is None:
                    if order < zech_log_bound:
                        # DO *NOT* use for prime subfield, since that would lead to
                        # a circular reference in the call to ParentWithGens in the
                        # __init__ method.
                        impl = 'givaro'
                    elif order % 2 == 0:
                        impl = 'ntl'
                    else:
                        impl = 'pari_ffelt'
                if impl == 'givaro':
                    if kwds.has_key('repr'):
                        repr = kwds['repr']
                    else:
                        repr = 'poly'
                    K = FiniteField_givaro(order, name, modulus, repr=repr, cache=elem_cache)
                elif impl == 'ntl':
                    from finite_field_ntl_gf2e import FiniteField_ntl_gf2e
                    K = FiniteField_ntl_gf2e(order, name, modulus)
                elif impl == 'pari_ffelt':
                    from finite_field_pari_ffelt import FiniteField_pari_ffelt
                    K = FiniteField_pari_ffelt(p, modulus, name)
                elif (impl == 'pari_mod'
                      or impl == 'pari'):    # for unpickling old pickles
                    from finite_field_ext_pari import FiniteField_ext_pari
                    K = FiniteField_ext_pari(order, name, modulus)
                else:
                    raise ValueError("no such finite field implementation: %s" % impl)

            # Temporary; see create_key_and_extra_args() above.
            if kwds.has_key('prefix'):
                K._prefix = kwds['prefix']

        return K

    def create_key_and_extra_args(self, order, name=None, modulus=None, names=None,
                                  impl=None, proof=None, **kwds):
        """
        EXAMPLES::

            sage: GF.create_key_and_extra_args(9, 'a')
            ((9, ('a',), x^2 + 2*x + 2, None, '{}', 3, 2, True), {})
            sage: GF.create_key_and_extra_args(9, 'a', foo='value')
            ((9, ('a',), x^2 + 2*x + 2, None, "{'foo': 'value'}", 3, 2, True), {'foo': 'value'})
        """
        from sage.structure.proof.all import WithProof, arithmetic
        if proof is None: proof = arithmetic()
        with WithProof('arithmetic', proof):
            order = int(order)
            if order <= 1:
                raise ValueError("the order of a finite field must be > 1.")

            if arith.is_prime(order):
                name = None
                modulus = None
                p = integer.Integer(order)
                n = integer.Integer(1)
            elif arith.is_prime_power(order):
                if not names is None: name = names
                name = normalize_names(1,name)

                p, n = arith.factor(order)[0]

                # The following is a temporary solution that allows us
                # to construct compatible systems of finite fields
                # until algebraic closures of finite fields are
                # implemented in Sage.  It requires the user to
                # specify two parameters:
                #
                # - `conway` -- boolean; if True, this field is
                #   constructed to fit in a compatible system using
                #   a Conway polynomial.
                # - `prefix` -- a string used to generate names for
                #   automatically constructed finite fields
                #
                # See the docstring of FiniteFieldFactory for examples.
                #
                # Once algebraic closures of finite fields are
                # implemented, this syntax should be superseded by
                # something like the following:
                #
                #     sage: Fpbar = GF(5).algebraic_closure('z')
                #     sage: F, e = Fpbar.subfield(3)  # e is the embedding into Fpbar
                #     sage: F
                #     Finite field in z3 of size 5^3
                #
                # This temporary solution only uses actual Conway
                # polynomials (no pseudo-Conway polynomials), since
                # pseudo-Conway polynomials are not unique, and until
                # we have algebraic closures of finite fields, there
                # is no good place to store a specific choice of
                # pseudo-Conway polynomials.
                if name is None:
                    if not (kwds.has_key('conway') and kwds['conway']):
                        raise ValueError("parameter 'conway' is required if no name given")
                    if not kwds.has_key('prefix'):
                        raise ValueError("parameter 'prefix' is required if no name given")
                    name = kwds['prefix'] + str(n)

                if kwds.has_key('conway') and kwds['conway']:
                    from conway_polynomials import conway_polynomial
                    if not kwds.has_key('prefix'):
                        raise ValueError("a prefix must be specified if conway=True")
                    if modulus is not None:
                        raise ValueError("no modulus may be specified if conway=True")
                    # The following raises a RuntimeError if no polynomial is found.
                    modulus = conway_polynomial(p, n)

                if modulus is None or isinstance(modulus, str):
                    # A string specifies an algorithm to find a suitable modulus.
                    if modulus == "default":    # for backward compatibility
                        modulus = None
                    modulus = GF(p)['x'].irreducible_element(n, algorithm=modulus)
                elif isinstance(modulus, (list, tuple)):
                    modulus = GF(p)['x'](modulus)
                elif sage.rings.polynomial.polynomial_element.is_Polynomial(modulus):
                    modulus = modulus.change_variable_name('x')
                else:
                    raise TypeError("wrong type for modulus parameter")
            else:
                raise ValueError("the order of a finite field must be a prime power.")

            return (order, name, modulus, impl, str(kwds), p, n, proof), kwds

#.........这里部分代码省略.........
        if existence:
            return True
        elif explain_construction:
            return "The database contains a ({},{},{})-difference family".format(v,k,l)

        vv, blocks = next(DF[v,k,l].iteritems())

        # Build the group
        from sage.rings.finite_rings.integer_mod_ring import Zmod
        if len(vv) == 1:
            G = Zmod(vv[0])
        else:
            from sage.categories.cartesian_product import cartesian_product
            G = cartesian_product([Zmod(i) for i in vv])

        df = [[G(i) for i in b] for b in blocks]

        if check and not is_difference_family(G, df, v=v, k=k, l=l):
            raise RuntimeError("There is an invalid ({},{},{})-difference "
                    "family in the database... Please contact "
                    "[email protected]".format(v,k,l))

        return G,df

    e = k*(k-1)
    if (l*(v-1)) % e:
        if existence:
            return Unknown
        raise NotImplementedError("No construction available for ({},{},{})-difference family".format(v,k,l))
    t = l*(v-1) // e  # number of blocks

    # trivial construction
    if k == (v-1) and l == (v-2):
        from sage.rings.finite_rings.integer_mod_ring import Zmod
        G = Zmod(v)
        return G, [range(1,v)]

    factorization = arith.factor(v)
    D = None

    if len(factorization) == 1:  # i.e. is v a prime power
        from sage.rings.finite_rings.constructor import GF
        G = K = GF(v,'z')

        if radical_difference_family(K, k, l, existence=True):
            if existence:
                return True
            elif explain_construction:
                return "Radical difference family on a finite field"
            else:
                D = radical_difference_family(K,k,l)

        elif l == 1 and k == 6 and df_q_6_1(K,existence=True):
            if existence:
                return True
            elif explain_construction:
                return "Wilson 1972 difference family made from the union of two cyclotomic cosets"
            else:
                D = df_q_6_1(K)

    # Twin prime powers construction
    # i.e. v = p(p+2) where p and p+2 are prime powers
    #      k = (v-1)/2
    #      lambda = (k-1)/2 (ie 2l+1 = k)
    elif (k == (v-1)//2 and
          l == (k-1)//2 and
          len(factorization) == 2 and
          abs(pow(*factorization[0]) - pow(*factorization[1])) == 2):
        if existence:
            return True