Python arith.valuation函数代码示例

def local_density(self, p, m):
    """
    Gives the local density -- should be called by the user. =)

    NOTE: This screens for imprimitive forms, and puts the quadratic
    form in local normal form, which is a *requirement* of the
    routines performing the computations!

    INPUT:

        `p` -- a prime number > 0
        `m` -- an integer

    OUTPUT:

        a rational number

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])   ## NOTE: This is already in local normal form for *all* primes p!
        sage: Q.local_density(p=2, m=1)
        1
        sage: Q.local_density(p=3, m=1)
        8/9
        sage: Q.local_density(p=5, m=1)
        24/25
        sage: Q.local_density(p=7, m=1)
        48/49
        sage: Q.local_density(p=11, m=1)
        120/121

    """
    n = self.dim()
    if (n == 0):
        raise TypeError("Oops!  We currently don't handle 0-dim'l forms. =(")

    ## Find the local normal form and p-scale of Q     --  Note: This uses the valuation ordering of local_normal_form.
    ##                                                     TO DO:  Write a separate p-scale and p-norm routines!
    Q_local = self.local_normal_form(p)
    if n == 1:
        p_valuation = valuation(Q_local[0,0], p)
    else:
        p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p))

    ## If m is less p-divisible than the matrix, return zero
    if ((m != 0) and (valuation(m,p) < p_valuation)):   ## Note: The (m != 0) condition protects taking the valuation of zero.
        return QQ(0)


    ## If the form is imprimitive, rescale it and call the local density routine
    p_adjustment = QQ(1) / p**p_valuation
    m_prim = QQ(m) / p**p_valuation
    Q_prim = Q_local.scale_by_factor(p_adjustment)

    ## Return the densities for the reduced problem
    return Q_prim.local_density_congruence(p, m_prim)

def find_entry_with_minimal_scale_at_prime(self, p):
    """
    Finds the entry of the quadratic form with minimal scale at the
    prime p, preferring diagonal entries in case of a tie.  (I.e.  If
    we write the quadratic form as a symmetric matrix M, then this
    entry M[i,j] has the minimal valuation at the prime p.)

    Note: This answer is independent of the kind of matrix (Gram or
    Hessian) associated to the form.

    INPUT:

        `p` -- a prime number > 0

    OUTPUT:

        a pair of integers >= 0

    EXAMPLES::

        sage: Q = QuadraticForm(ZZ, 2, [6, 2, 20]); Q
        Quadratic form in 2 variables over Integer Ring with coefficients:
        [ 6 2 ]
        [ * 20 ]
        sage: Q.find_entry_with_minimal_scale_at_prime(2)
        (0, 1)
        sage: Q.find_entry_with_minimal_scale_at_prime(3)
        (1, 1)
        sage: Q.find_entry_with_minimal_scale_at_prime(5)
        (0, 0)

    """
    n = self.dim()
    min_val = Infinity
    ij_index = None
    val_2 = valuation(2, p)
    for d in range(n):           ## d = difference j-i
        for e in range(n - d):    ## e is the length of the diagonal with value d.

            ## Compute the valuation of the entry
            if d == 0:
                tmp_val = valuation(self[e, e+d], p)
            else:
                tmp_val = valuation(self[e, e+d], p) - val_2

            ## Check if it's any smaller than what we have
            if tmp_val < min_val:
                ij_index = (e,e+d)
                min_val = tmp_val

    ## Return the result
    return ij_index

    def _push(self, x):
        f = self._t
        level = valuation(x.parent().degree(), self._degree)

        p = [self[level-1](list(c))
             for c in izip_longest(*decompose(x.lift(), f), fillvalue=0)]
        return p if p else [self[level-1](0)]

 def _e_bounds(self, n, prec):
     p = self._p
     prec = max(2,prec)
     R = PowerSeriesRing(ZZ,'T',prec+1)
     T = R(R.gen(),prec +1)
     w = (1+T)**(p**n) - 1
     return [infinity] + [valuation(w[j],p) for j in range(1,min(w.degree()+1,prec))]

    def _push(self, x):
        level = valuation(x.parent().degree(), self._degree)
        f, g = self._rel_polys[-level % len(self._rel_polys)]
        deg = self._degree**(level - 1) - 1
        x *= x.parent(g**deg)

        p = [self[level-1](list(c))
             for c in izip_longest(*decompose(x.lift(), f, g, deg), fillvalue=0)]
        return p if p else [self[level-1](0)]

    def _lift(self, xs):
        if not xs:
            raise RuntimeError("Don't know where to lift to.")

        f = self._t
        Ps = map(self._P.__call__, izip_longest(*xs))
        level = valuation(xs[0].parent().degree(), self._degree)

        return self[level+1](compose(Ps, f))

    def _lift(self, xs):
        if not xs:
            raise RuntimeError("Don't know where to lift to.")

        level = valuation(xs[0].parent().degree(), self._degree)
        f, g = self._rel_polys[(-level-1) % len(self._rel_polys)]
        Ps = map(self._P.__call__, izip_longest(*xs))

        return (self[level+1](compose(Ps, f, g)) / 
                self[level+1](g**(len(Ps)-1)))

	def valuation(self,p):
		"""returns the exponent of the highest power of p which divides all coefficients of self"""
		#assert self.base_ring==QQ, "need to be working over Q in valuation"
		k=self.weight
		v=self.vars()
		X=v[0]
		Y=v[1]
		v=[]
		for j in range(k+1):
			v=v+[valuation(QQ(self.poly.coefficient((X**j)*(Y**(k-j)))),p)]
		return min(v)

def local_normal_form(self, p):
    """
    Returns the a locally integrally equivalent quadratic form over
    the p-adic integers Z_p which gives the Jordan decomposition.  The
    Jordan components are written as sums of blocks of size <= 2 and
    are arranged by increasing scale, and then by increasing norm.
    (This is equivalent to saying that we put the 1x1 blocks before
    the 2x2 blocks in each Jordan component.)

    INPUT:
        `p` -- a positive prime number.

    OUTPUT:
        a quadratic form over ZZ
    
    WARNING:  Currently this only works for quadratic forms defined over ZZ.
    
    EXAMPLES::
    
        sage: Q = QuadraticForm(ZZ, 2, [10,4,1])
        sage: Q.local_normal_form(5)
        Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 1 0 ]
        [ * 6 ]
        
    ::            
    
        sage: Q.local_normal_form(3)
        Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 10 0 ]
        [ * 15 ]
    
        sage: Q.local_normal_form(2)
        Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 1 0 ]
        [ * 6 ]
    
    """
    ## Sanity Checks
    if (self.base_ring() != IntegerRing()):
        raise NotImplementedError, "Oops!  This currently only works for quadratic forms defined over IntegerRing(). =("
    if not ((p>=2) and is_prime(p)):
        raise TypeError, "Oops!  p is not a positive prime number. =("

    ## Some useful local variables
    Q = copy.deepcopy(self)
    Q.__init__(self.base_ring(), self.dim(), self.coefficients())

    ## Prepare the final form to return
    Q_Jordan = copy.deepcopy(self)
    Q_Jordan.__init__(self.base_ring(), 0)

    
    while Q.dim() > 0:
        n = Q.dim()

        ## Step 1: Find the minimally p-divisible matrix entry, preferring diagonals
        ## -------------------------------------------------------------------------
        (min_i, min_j) = Q.find_entry_with_minimal_scale_at_prime(p)
        if min_i == min_j:
            min_val = valuation(2 * Q[min_i, min_j], p)
        else:
            min_val = valuation(Q[min_i, min_j], p)

        ## Error if we still haven't seen non-zero coefficients!
        if (min_val == Infinity):
            raise RuntimeError, "Oops!  The original matrix is degenerate. =("

      
        ## Step 2: Arrange for the upper leftmost entry to have minimal valuation
        ## ----------------------------------------------------------------------
        if (min_i == min_j):
            block_size = 1
            Q.swap_variables(0, min_i, in_place = True)
        else:
            ## Work in the upper-left 2x2 block, and replace it by its 2-adic equivalent form  
            Q.swap_variables(0, min_i, in_place = True)
            Q.swap_variables(1, min_j, in_place = True)

            ## 1x1 => make upper left the smallest
            if (p != 2):
                block_size = 1;
                Q.add_symmetric(1, 0, 1, in_place = True)   
            ## 2x2 => replace it with the appropriate 2x2 matrix
            else: 
                block_size = 2

        ## DIAGNOSTIC
        #print "\n Finished Step 2 \n";
        #print "\n Q is: \n" + str(Q)  + "\n";
        #print "  p is: " + str(p) 
        #print "  min_val is: " + str( min_val)
        #print "  block_size is: " + str(block_size)
        #print "\n Starting Step 3 \n"

        ## Step 3: Clear out the remaining entries
        ##  ---------------------------------------
        min_scale = p ** min_val                             ## This is the minimal valuation of the Hessian matrix entries.

        ##DIAGNOSTIC
#.........这里部分代码省略.........

def jordan_blocks_by_scale_and_unimodular(self, p, safe_flag=True):
    """
    Returns a list of pairs `(s_i, L_i)` where `L_i` is a maximal
    `p^{s_i}`-unimodular Jordan component which is further decomposed into
    block diagonals of block size `\le 2`. For each `L_i` the 2x2 blocks are
    listed after the 1x1 blocks (which follows from the convention of the
    :meth:`local_normal_form` method).

    ..note ::
    
        The decomposition of each `L_i` into smaller block is not unique!

    The ``safe_flag`` argument allows us to select whether we want a copy of
    the output, or the original output.  By default ``safe_flag = True``, so we
    return a copy of the cached information.  If this is set to ``False``, then
    the routine is much faster but the return values are vulnerable to being
    corrupted by the user. 

    INPUT:

    - `p` -- a prime number > 0.

    OUTPUT:

    A list of pairs `(s_i, L_i)` where: 
    
    - `s_i` is an integer, 
    - `L_i` is a block-diagonal unimodular quadratic form over `\ZZ_p`.
     
    .. note::
    
        These forms `L_i` are defined over the `p`-adic integers, but by a
        matrix over `\ZZ` (or `\QQ`?).

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7])
        sage: Q.jordan_blocks_by_scale_and_unimodular(3)
        [(0, Quadratic form in 3 variables over Integer Ring with coefficients: 
        [ 1 0 0 ]
        [ * 5 0 ]
        [ * * 7 ]), (2, Quadratic form in 1 variables over Integer Ring with coefficients: 
        [ 1 ])]
        
    ::

        sage: Q2 = QuadraticForm(ZZ, 2, [1,1,1])
        sage: Q2.jordan_blocks_by_scale_and_unimodular(2)
        [(-1, Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 2 2 ]
        [ * 2 ])]
        sage: Q = Q2 + Q2.scale_by_factor(2)
        sage: Q.jordan_blocks_by_scale_and_unimodular(2)
        [(-1, Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 2 2 ]
        [ * 2 ]), (0, Quadratic form in 2 variables over Integer Ring with coefficients: 
        [ 2 2 ]
        [ * 2 ])]
    """
    ## Try to use the cached result
    try:
        if safe_flag:
            return copy.deepcopy(self.__jordan_blocks_by_scale_and_unimodular_dict[p])
        else:
            return self.__jordan_blocks_by_scale_and_unimodular_dict[p]
    except StandardError:
        ## Initialize the global dictionary if it doesn't exist
        if not hasattr(self, '__jordan_blocks_by_scale_and_unimodular_dict'):
            self.__jordan_blocks_by_scale_and_unimodular_dict = {}


    ## Deal with zero dim'l forms
    if self.dim() == 0:
        return []


    ## Find the Local Normal form of Q at p
    Q1 = self.local_normal_form(p)


    ## Parse this into Jordan Blocks
    n = Q1.dim()
    tmp_Jordan_list = []
    i = 0
    start_ind = 0
    if (n >= 2) and (Q1[0,1] != 0):
        start_scale = valuation(Q1[0,1], p) - 1
    else:
        start_scale = valuation(Q1[0,0], p)    

    while (i < n):

        ## Determine the size of the current block
        if (i == n-1) or (Q1[i,i+1] == 0):
            block_size = 1
        else: 
            block_size = 2

        ## Determine the valuation of the current block
        if block_size == 1:
#.........这里部分代码省略.........

def local_badII_density_congruence(self, p, m, Zvec=None, NZvec=None):
    """
    Finds the Bad-type II local density of Q representing `m` at `p`.
    (Assuming that `p` > 2 and Q is given in local diagonal form.)


     INPUT:

        Q -- quadratic form assumed to be block diagonal and p-integral

        `p` -- a prime number

        `m` -- an integer

        Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None

    OUTPUT:

        a rational number

    EXAMPLES::

        sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3])
        sage: Q.local_badII_density_congruence(2, 1, None, None)
        0
        sage: Q.local_badII_density_congruence(2, 2, None, None)
        0
        sage: Q.local_badII_density_congruence(2, 4, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 1, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 6, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 9, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 27, None, None)
        0

    ::

        sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9])
        sage: Q.local_badII_density_congruence(3, 1, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 3, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 6, None, None)
        0
        sage: Q.local_badII_density_congruence(3, 9, None, None)
        4/27
        sage: Q.local_badII_density_congruence(3, 18, None, None)
        4/9

    """
    ## DIAGNOSTIC
    verbose(" In local_badII_density_congruence with ")
    verbose(" Q is: \n" + str(self))
    verbose(" p = " + str(p))
    verbose(" m = " + str(m))
    verbose(" Zvec = " + str(Zvec))
    verbose(" NZvec = " + str(NZvec))

    ## Put the Zvec congruence condition in a standard form
    if Zvec is None:
        Zvec = []


    n = self.dim()


    ## Sanity Check on Zvec and NZvec:
    ## -------------------------------
    Sn = Set(range(n))
    if (Zvec is not None) and (len(Set(Zvec) + Sn) > n):
        raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.")
    if (NZvec is not None) and (len(Set(NZvec) + Sn) > n):
        raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.")


    ## Define the indexing sets S_i:
    ## -----------------------------
    S0 = []
    S1 = []
    S2plus = []

    for i in range(n):

        ## Compute the valuation of each index, allowing for off-diagonal terms
        if (self[i,i] == 0):
            if (i == 0):
                val = valuation(self[i,i+1], p)    ## Look at the term to the right
            elif (i == n-1):
                val = valuation(self[i-1,i], p)    ## Look at the term above
            else:
                val = valuation(self[i,i+1] + self[i-1,i], p)    ## Finds the valuation of the off-diagonal term since only one isn't zero
        else:
            val = valuation(self[i,i], p)

        ## Sort the indices into disjoint sets by their valuation
        if (val == 0):
            S0 += [i]
#.........这里部分代码省略.........

#.........这里部分代码省略.........

    ## Put the Zvec congruence condition in a standard form
    if Zvec is None:
        Zvec = []


    n = self.dim()



    ## Sanity Check on Zvec and NZvec:
    ## -------------------------------
    Sn = Set(range(n))
    if (Zvec is not None) and (len(Set(Zvec) + Sn) > n):
        raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.")
    if (NZvec is not None) and (len(Set(NZvec) + Sn) > n):
        raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.")



    ## Define the indexing set S_0, and determine if S_1 is empty:
    ## -----------------------------------------------------------
    S0 = []
    S1_empty_flag = True    ## This is used to check if we should be computing BI solutions at all!
                            ## (We should really to this earlier, but S1 must be non-zero to proceed.)

    ## Find the valuation of each variable (which will be the same over 2x2 blocks),
    ## remembering those of valuation 0 and if an entry of valuation 1 exists.
    for i in range(n):

        ## Compute the valuation of each index, allowing for off-diagonal terms
        if (self[i,i] == 0):
            if (i == 0):
                val = valuation(self[i,i+1], p)    ## Look at the term to the right
            else:
                if (i == n-1):
                    val = valuation(self[i-1,i], p)    ## Look at the term above
                else:
                    val = valuation(self[i,i+1] + self[i-1,i], p)    ## Finds the valuation of the off-diagonal term since only one isn't zero
        else:
            val = valuation(self[i,i], p)

        if (val == 0):
            S0 += [i]
        elif (val == 1):
            S1_empty_flag = False    ## Need to have a non-empty S1 set to proceed with Bad-type I reduction...





    ## Check that S1 is non-empty and p|m to proceed, otherwise return no solutions.
    if (S1_empty_flag == True) or (m % p != 0):
        return 0

    ## Check some conditions for no bad-type I solutions to exist
    if (NZvec is not None) and (len(Set(S0).intersection(Set(NZvec))) != 0):
        return 0



    ## Check that the form is primitive...                     WHY DO WE NEED TO DO THIS?!?
    if (S0 == []):
        print " Using Q = " + str(self)
        print " and p = " + str(p)
        raise RuntimeError("Oops! The form is not primitive!")

#.........这里部分代码省略.........
            1
            sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=True)
            sage: m._scaling
            -1
            sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=True)
            sage: m._scaling
            -1/2
            sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=False)
            sage: m._scaling
            2
            sage: m = EllipticCurve('196a1').modular_symbol(use_eclib=False)
            sage: m._scaling  
            1/2  
            
        Some harder cases fail::
        
            sage: m = EllipticCurve('121b1').modular_symbol(use_eclib=False)
            Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
            sage: m._scaling  
            1           
            
        TESTS::

            sage: rk0 = ['11a1', '11a2', '15a1', '27a1', '37b1']
            sage: for la in rk0:  # long time (3s on sage.math, 2011)
            ...          E = EllipticCurve(la)
            ...          me = E.modular_symbol(use_eclib = True)
            ...          ms = E.modular_symbol(use_eclib = False)
            ...          print E.lseries().L_ratio()*E.real_components(), me(0), ms(0)
            1/5 1/5 1/5
            1 1 1
            1/4 1/4 1/4
            1/3 1/3 1/3
            2/3 2/3 2/3

            sage: rk1 = ['37a1','43a1','53a1', '91b1','91b2','91b3']
            sage: [EllipticCurve(la).modular_symbol(use_eclib=True)(0) for la in rk1]  # long time (1s on sage.math, 2011)
            [0, 0, 0, 0, 0, 0]
            sage: for la in rk1:  # long time (8s on sage.math, 2011)
            ...       E = EllipticCurve(la)
            ...       m = E.modular_symbol(use_eclib = True)
            ...       lp = E.padic_lseries(5)
            ...       for D in [5,17,12,8]:
            ...           ED = E.quadratic_twist(D)
            ...           md = sum([kronecker(D,u)*m(ZZ(u)/D) for u in range(D)])
            ...           etaa = lp._quotient_of_periods_to_twist(D)
            ...           assert ED.lseries().L_ratio()*ED.real_components()*etaa == md

        """
        E = self._E
        self._scaling = 1 # by now.
        self._failed_to_scale = False

        if self._sign == 1 :
            at0 = self(0)
            # print 'modular symbol evaluates to ',at0,' at 0'
            if at0 != 0 :
                l1 = self.__lalg__(1)
                if at0 != l1:
                    verbose('scale modular symbols by %s'%(l1/at0))
                    self._scaling = l1/at0
            else :
                # if [0] = 0, we can still hope to scale it correctly by considering twists of E
                Dlist = [5,8,12,13,17,21,24,28,29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97]  # a list of positive fundamental discriminants
                j = 0
                at0 = 0
                # computes [0]+ for the twist of E by D until one value is non-zero
                while j < 30 and at0 == 0 :
                    D = Dlist[j]
                    # the following line checks if the twist of the newform of E by D is a newform
                    # this is to avoid that we 'twist back'
                    if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
                        at0 = sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
                    j += 1
                if j == 30 and at0 == 0: # curves like "121b1", "225a1", "225e1", "256a1", "256b1", "289a1", "361a1", "400a1", "400c1", "400h1", "441b1", "441c1", "441d1", "441f1 .. will arrive here
                    self.__scale_by_periods_only__()
                else :
                    l1 = self.__lalg__(D)
                    if at0 != l1:
                        verbose('scale modular symbols by %s found at D=%s '%(l1/at0,D), level=2)
                        self._scaling = l1/at0

        else : # that is when sign = -1
            Dlist = [-3,-4,-7,-8,-11,-15,-19,-20,-23,-24, -31, -35, -39, -40, -43, -47, -51, -52, -55, -56, -59, -67, -68, -71, -79, -83, -84, -87, -88, -91]  # a list of negative fundamental discriminants
            j = 0
            at0 = 0
            while j < 30 and at0 == 0 :
                # computes [0]+ for the twist of E by D until one value is non-zero
                D = Dlist[j]
                if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
                    at0 = - sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
                j += 1
            if j == 30 and at0 == 0: # no more hope for a normalization
                # we do at least a scaling with the quotient of the periods
                self.__scale_by_periods_only__()
            else :
                l1 = self.__lalg__(D)
                if at0 != l1:
                    verbose('scale modular symbols by %s'%(l1/at0))
                    self._scaling = l1/at0

def local_primitive_density(self, p, m):
    """
    Gives the local primitive density -- should be called by the user. =)

    NOTE: This screens for imprimitive forms, and puts the
    quadratic form in local normal form, which is a *requirement* of
    the routines performing the computations!

    INPUT:
        `p` -- a prime number > 0
        `m` -- an integer

    OUTPUT:
        a rational number

    EXAMPLES::

        sage: Q = QuadraticForm(ZZ, 4, range(10))
        sage: Q[0,0] = 5
        sage: Q[1,1] = 10
        sage: Q[2,2] = 15
        sage: Q[3,3] = 20
        sage: Q
        Quadratic form in 4 variables over Integer Ring with coefficients:
        [ 5 1 2 3 ]
        [ * 10 5 6 ]
        [ * * 15 8 ]
        [ * * * 20 ]
        sage: Q.theta_series(20)
        1 + 2*q^5 + 2*q^10 + 2*q^14 + 2*q^15 + 2*q^16 + 2*q^18 + O(q^20)
        sage: Q.local_normal_form(2)
        Quadratic form in 4 variables over Integer Ring with coefficients:
        [ 0 1 0 0 ]
        [ * 0 0 0 ]
        [ * * 0 1 ]
        [ * * * 0 ]

        sage: Q.local_primitive_density(2, 1)
        3/4
        sage: Q.local_primitive_density(5, 1)
        24/25

        sage: Q.local_primitive_density(2, 5)
        3/4
        sage: Q.local_density(2, 5)
        3/4

    """
    n = self.dim()
    if (n == 0):
        raise TypeError("Oops!  We currently don't handle 0-dim'l forms. =(")

    ## Find the local normal form and p-scale of Q     --  Note: This uses the valuation ordering of local_normal_form.
    ##                                                     TO DO:  Write a separate p-scale and p-norm routines!
    Q_local = self.local_normal_form(p)
    if n == 1:
        p_valuation = valuation(Q_local[0,0], p)
    else:
        p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p))


    ## If m is less p-divisible than the matrix, return zero
    if ((m != 0) and (valuation(m,p) < p_valuation)):   ## Note: The (m != 0) condition protects taking the valuation of zero.
        return QQ(0)


    ## If the form is imprimitive, rescale it and call the local density routine
    p_adjustment = QQ(1) / p**p_valuation
    m_prim = QQ(m) / p**p_valuation
    Q_prim = Q_local.scale_by_factor(p_adjustment)

    ## Return the densities for the reduced problem
    return Q_prim.local_primitive_density_congruence(p, m_prim)

def has_equivalent_Jordan_decomposition_at_prime(self, other, p):
    """
    Determines if the given quadratic form has a Jordan decomposition
    equivalent to that of self.
    
    INPUT:
        a QuadraticForm

    OUTPUT:
        boolean
    
    EXAMPLES::

        sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 1, 0, 3])
        sage: Q2 = QuadraticForm(ZZ, 3, [1, 0, 0, 2, -2, 6])
        sage: Q3 = QuadraticForm(ZZ, 3, [1, 0, 0, 1, 0, 11])
        sage: [Q1.level(), Q2.level(), Q3.level()]
        [44, 44, 44]
        sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,2)
        False
        sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,11)
        False
        sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
        False
        sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
        True 
        sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
        True 
        sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
        False
    
    """
    ## Sanity Checks
    #if not isinstance(other, QuadraticForm):
    if type(other) != type(self):
        raise TypeError, "Oops!  The first argument must be of type QuadraticForm."
    if not is_prime(p):
        raise TypeError, "Oops!  The second argument must be a prime number."

    ## Get the relevant local normal forms quickly
    self_jordan = self.jordan_blocks_by_scale_and_unimodular(p, safe_flag= False)
    other_jordan = other.jordan_blocks_by_scale_and_unimodular(p, safe_flag=False)    

    ## DIAGNOSTIC
    #print "self_jordan = ", self_jordan
    #print "other_jordan = ", other_jordan


    ## Check for the same number of Jordan components
    if len(self_jordan) != len(other_jordan):
        return False


    ## Deal with odd primes:  Check that the Jordan component scales, dimensions, and discriminants are the same
    if p != 2:
        for i in range(len(self_jordan)):
            if (self_jordan[i][0] != other_jordan[i][0]) \
               or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
               or (legendre_symbol(self_jordan[i][1].det() * other_jordan[i][1].det(), p) != 1):
                return False

        ## All tests passed for an odd prime.
        return True


    ## For p = 2:  Check that all Jordan Invariants are the same.
    elif p == 2:

        ## Useful definition
        t = len(self_jordan)          ## Define t = Number of Jordan components


        ## Check that all Jordan Invariants are the same (scale, dim, and norm)
        for i in range(t):
            if (self_jordan[i][0] != other_jordan[i][0]) \
               or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
               or (valuation(GCD(self_jordan[i][1].coefficients()), p) != valuation(GCD(other_jordan[i][1].coefficients()), p)):
                return False

        ## DIAGNOSTIC
        #print "Passed the Jordan invariant test."


        ## Use O'Meara's isometry test 93:29 on p277.
        ## ------------------------------------------

        ## List of norms, scales, and dimensions for each i
        scale_list = [ZZ(2)**self_jordan[i][0]  for i in range(t)]
        norm_list = [ZZ(2)**(self_jordan[i][0] + valuation(GCD(self_jordan[i][1].coefficients()), 2))  for i in range(t)]
        dim_list = [(self_jordan[i][1].dim())  for i in range(t)]

        ## List of Hessian determinants and Hasse invariants for each Jordan (sub)chain
        ## (Note: This is not the same as O'Meara's Gram determinants, but ratios are the same!)  -- NOT SO GOOD...
        ## But it matters in condition (ii), so we multiply all by 2 (instead of dividing by 2 since only square-factors matter, and it's easier.)
        j = 0
        self_chain_det_list = [ self_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])]
        other_chain_det_list = [ other_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])]
        self_hasse_chain_list = [ self_jordan[j][1].scale_by_factor(ZZ(2)**self_jordan[j][0]).hasse_invariant__OMeara(2) ]
        other_hasse_chain_list = [ other_jordan[j][1].scale_by_factor(ZZ(2)**other_jordan[j][0]).hasse_invariant__OMeara(2) ]
                   
#.........这里部分代码省略.........