def __iter__(self):
"""
Return an iterator over the elements of this group.
EXAMPLES::
sage: G = AbelianGroup([2,3], names = "ab")
sage: [a for a in G]
[1, b, b^2, a, a*b, a*b^2]
sage: L = list(G); L
[1, b, b^2, a, a*b, a*b^2]
The returned list is a reference; mutating it does not allow the
user to (accidentally?) alter the computed generators::
sage: L[0] = 0
sage: list(G)
[1, b, b^2, a, a*b, a*b^2]
sage: G = AbelianGroup([1], names="a")
sage: list(G)
[1]
sage: G = AbelianGroup([])
sage: G.list()
[1]
sage: list(G)
[1]
"""
invs = self.invariants()
for t in mrange(invs):
yield AbelianGroupElement(self, t)
def __iter__(self):
"""
Return an iterator over the elements of this group.
EXAMPLES::
sage: G = AbelianGroup([2,3], names = "ab")
sage: [a for a in G]
[1, b, b^2, a, a*b, a*b^2]
sage: L = list(G); L
[1, b, b^2, a, a*b, a*b^2]
The returned list is a reference; mutating it does not allow the
user to (accidentally?) alter the computed generators::
sage: L[0] = 0
sage: list(G)
[1, b, b^2, a, a*b, a*b^2]
sage: G = AbelianGroup([1], names="a")
sage: list(G)
[1]
sage: G = AbelianGroup([])
sage: G.list()
[1]
sage: list(G)
[1]
"""
invs = self.invariants()
for t in mrange(invs):
yield AbelianGroupElement(self, t)
def mass_at_two_by_counting_mod_power(self, k):
"""
Computes the local mass at `p=2` assuming that it's stable `(mod 2^k)`.
Note: This is **way** too slow to be useful, even when k=1!!!
TO DO: Remove this routine, or try to compile it!
INPUT:
k -- an integer >= 1
OUTPUT:
a rational number
EXAMPLE::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass_at_two_by_counting_mod_power(1)
4
"""
R = IntegerModRing(2**k)
Q1 = self.base_change_to(R)
n = self.dim()
MS = MatrixSpace(R, n)
ct = sum([1 for x in mrange([2**k] * (n**2)) if Q1(MS(x)) == Q1]) ## Count the solutions mod 2^k
two_mass = ZZ(1)/2 * (ZZ(ct) / ZZ(2)**(k*n*(n-1)/2))
return two_mass
def mass_at_two_by_counting_mod_power(self, k):
"""
Computes the local mass at `p=2` assuming that it's stable `(mod 2^k)`.
Note: This is **way** too slow to be useful, even when k=1!!!
TO DO: Remove this routine, or try to compile it!
INPUT:
k -- an integer >= 1
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass_at_two_by_counting_mod_power(1)
4
"""
R = IntegerModRing(2**k)
Q1 = self.base_change_to(R)
n = self.dim()
MS = MatrixSpace(R, n)
ct = sum([1 for x in mrange([2**k] * (n**2))
if Q1(MS(x)) == Q1]) ## Count the solutions mod 2^k
two_mass = ZZ(1) / 2 * (ZZ(ct) / ZZ(2)**(k * n * (n - 1) / 2))
return two_mass
def dual_vectors( self):
"""
Return a set of representatives
for $L^#/L$.
"""
D,U,V = self.gram_matrix().smith_form()
# hence D = U * self.gram_matrix() * V
if None == self.__dual_vectors:
W = V*D**-1
S = self.space()
self.__dual_vectors = [ W*S(v) for v in mrange( D.diagonal())]
return self.__dual_vectors
def cs_range( f, subset = None):
"""
For a symmetric semi-positive integral matrix $f$,
return a list of all integral $n$-vectors $v$ such that
$x^tfx - (v*x)^2 >= 0$ for all $x$.
"""
n = f.dimensions()[0]
b = vector( s.isqrt() for s in f.diagonal())
zv = vector([0]*n)
box = [b - vector(t) for t in mrange( (2*b + vector([1]*n)).list()) if b - vector(t) > zv]
if subset:
box = [v for v in box if v in subset]
rge = [v for v in box if min( (f - matrix(v).transpose()*matrix(v)).eigenvalues()) >=0]
return rge
def list(self):
"""
Return tuple of all elements of this group.
EXAMPLES::
sage: G = AbelianGroup([2,3], names="ab")
sage: Gd = G.dual_group(names="AB")
sage: Gd.list()
(1, B, B^2, A, A*B, A*B^2)
"""
if not (self.is_finite()):
raise NotImplementedError("Group must be finite")
invs = self.gens_orders()
T = mrange(invs)
n = self.order()
L = tuple(self(t) for t in T)
return L
def list(self):
"""
Return tuple of all elements of this group.
EXAMPLES::
sage: G = AbelianGroup([2,3], names="ab")
sage: Gd = G.dual_group(names="AB")
sage: Gd.list()
(1, B, B^2, A, A*B, A*B^2)
"""
if not(self.is_finite()):
raise NotImplementedError("Group must be finite")
invs = self.gens_orders()
T = mrange(invs)
n = self.order()
L = tuple( self(t) for t in T )
return L
def __iter__(self):
r"""
Return an iterator of all ideal classes in this class group.
EXAMPLES::
sage: K.<a> = NumberField(x^4 + 23)
sage: G = K.class_group()
sage: G
Class group of order 3 with structure C3 of Number Field
in a with defining polynomial x^4 + 23
sage: list(G)
[Trivial principal fractional ideal class,
Fractional ideal class (2, 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/4),
Fractional ideal class (2, 1/2*a^2 + 1/2)]
sage: G.list()
(Trivial principal fractional ideal class,
Fractional ideal class (2, 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/4),
Fractional ideal class (2, 1/2*a^2 + 1/2))
TESTS::
sage: K.<a> = NumberField(x^2 + 1)
sage: G = K.class_group()
sage: G
Class group of order 1 of Number Field in a with defining polynomial x^2 + 1
sage: list(G)
[Trivial principal fractional ideal class]
sage: G.list()
(Trivial principal fractional ideal class,)
"""
from sage.misc.mrange import mrange
orders = self.gens_orders()
T = mrange(orders)
g = self.gens()
for t in T:
I = self(1)
for i, j in enumerate(t):
I *= g[i]**j
yield I
if not T:
yield self(1)
def __iter__(self):
r"""
Return an iterator of all ideal classes in this class group.
EXAMPLES::
sage: K.<a> = NumberField(x^4 + 23)
sage: G = K.class_group()
sage: G
Class group of order 3 with structure C3 of Number Field
in a with defining polynomial x^4 + 23
sage: list(G)
[Trivial principal fractional ideal class,
Fractional ideal class (2, 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/4),
Fractional ideal class (2, 1/2*a^2 + 1/2)]
sage: G.list()
(Trivial principal fractional ideal class,
Fractional ideal class (2, 1/4*a^3 - 1/4*a^2 + 1/4*a - 1/4),
Fractional ideal class (2, 1/2*a^2 + 1/2))
TESTS::
sage: K.<a> = NumberField(x^2 + 1)
sage: G = K.class_group()
sage: G
Class group of order 1 of Number Field in a with defining polynomial x^2 + 1
sage: list(G)
[Trivial principal fractional ideal class]
sage: G.list()
(Trivial principal fractional ideal class,)
"""
from sage.misc.mrange import mrange
orders = self.gens_orders()
T = mrange(orders)
g = self.gens()
for t in T:
I = self(1)
for i, j in enumerate(t):
I *= g[i]**j
yield I
if not T:
yield self(1)
def list(self):
"""
Return list of all elements of this group.
EXAMPLES::
sage: G = AbelianGroup([2,3], names = "ab")
sage: Gd = DualAbelianGroup(G, names = "AB")
sage: Gd.list()
[1, B, B^2, A, A*B, A*B^2]
"""
try:
return list(self.__list)
except AttributeError:
pass
if not(self.is_finite()):
raise NotImplementedError, "Group must be finite"
invs = self.invariants()
T = mrange(invs)
n = self.order()
L = [DualAbelianGroupElement(self, t) for t in T]
self.__list = L
return list(self.__list)
def Watson_mass_at_2(self):
"""
Returns the local mass of the quadratic form when `p=2`, according
to Watson's Theorem 1 of "The 2-adic density of a quadratic form"
in Mathematika 23 (1976), pp 94--106.
INPUT:
none
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.Watson_mass_at_2() ## WARNING: WE NEED TO CHECK THIS CAREFULLY!
384
"""
## Make a 0-dim'l quadratic form (for initialization purposes)
Null_Form = copy.deepcopy(self)
Null_Form.__init__(ZZ, 0)
## Step 0: Compute Jordan blocks and bounds of the scales to keep track of
Jordan_Blocks = self.jordan_blocks_by_scale_and_unimodular(2)
scale_list = [B[0] for B in Jordan_Blocks]
s_min = min(scale_list)
s_max = max(scale_list)
## Step 1: Compute dictionaries of the diagonal block and 2x2 block for each scale
diag_dict = dict(
(i, Null_Form)
for i in range(s_min - 2, s_max + 4)) ## Initialize with the zero form
dim2_dict = dict(
(i, Null_Form)
for i in range(s_min, s_max + 4)) ## Initialize with the zero form
for (s, L) in Jordan_Blocks:
i = 0
while (i < L.dim() - 1) and (L[i, i + 1]
== 0): ## Find where the 2x2 blocks start
i = i + 1
if i < (L.dim() - 1):
diag_dict[s] = L.extract_variables(range(i)) ## Diagonal Form
dim2_dict[s + 1] = L.extract_variables(range(
i, L.dim())) ## Non-diagonal Form
else:
diag_dict[s] = L
#print "diag_dict = ", diag_dict
#print "dim2_dict = ", dim2_dict
#print "Jordan_Blocks = ", Jordan_Blocks
## Step 2: Compute three dictionaries of invariants (for n_j, m_j, nu_j)
n_dict = dict((j, 0) for j in range(s_min + 1, s_max + 2))
m_dict = dict((j, 0) for j in range(s_min, s_max + 4))
for (s, L) in Jordan_Blocks:
n_dict[s + 1] = L.dim()
if diag_dict[s].dim() == 0:
m_dict[s + 1] = ZZ.one() / ZZ(2) * L.dim()
else:
m_dict[s + 1] = floor(ZZ(L.dim() - 1) / ZZ(2))
#print " ==>", ZZ(L.dim() - 1) / ZZ(2), floor(ZZ(L.dim() - 1) / ZZ(2))
nu_dict = dict((j, n_dict[j + 1] - 2 * m_dict[j + 1])
for j in range(s_min, s_max + 1))
nu_dict[s_max + 1] = 0
#print "n_dict = ", n_dict
#print "m_dict = ", m_dict
#print "nu_dict = ", nu_dict
## Step 3: Compute the e_j dictionary
eps_dict = {}
for j in range(s_min, s_max + 3):
two_form = (diag_dict[j - 2] + diag_dict[j] +
dim2_dict[j]).scale_by_factor(2)
j_form = (two_form + diag_dict[j - 1]).base_change_to(
IntegerModRing(4))
if j_form.dim() == 0:
eps_dict[j] = 1
else:
iter_vec = [4] * j_form.dim()
alpha = sum([True for x in mrange(iter_vec) if j_form(x) == 0])
beta = sum([True for x in mrange(iter_vec) if j_form(x) == 2])
if alpha > beta:
eps_dict[j] = 1
elif alpha == beta:
eps_dict[j] = 0
else:
eps_dict[j] = -1
#print "eps_dict = ", eps_dict
## Step 4: Compute the quantities nu, q, P, E for the local mass at 2
nu = sum([j * n_dict[j] * (ZZ(n_dict[j] + 1) / ZZ(2) + \
sum([n_dict[r] for r in range(j+1, s_max+2)])) for j in range(s_min+1, s_max+2)])
q = sum([
sgn(nu_dict[j - 1] * (n_dict[j] + sgn(nu_dict[j])))
for j in range(s_min + 1, s_max + 2)
])
P = prod([
prod([1 - QQ(4)**(-i) for i in range(1, m_dict[j] + 1)])
for j in range(s_min + 1, s_max + 2)
])
E = prod([
ZZ(1) / ZZ(2) * (1 + eps_dict[j] * QQ(2)**(-m_dict[j]))
for j in range(s_min, s_max + 3)
])
#print "\nFinal Summary:"
#print "nu =", nu
#print "q = ", q
#print "P = ", P
#print "E = ", E
## Step 5: Compute the local mass for the prime 2.
mass_at_2 = QQ(2)**(nu - q) * P / E
return mass_at_2
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
#A1 = A.inverse() * A.det()
#Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
#Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
#ct = 0
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
#Q1 = self.matrix()
#if self(M) == self:
#ct += 1
#print "ct = ", ct, " M = "
#print M
#print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def minkowski_reduction_for_4vars__SP(self):
"""
Find a Minkowski-reduced form equivalent to the given one.
This means that
Q(`v_k`) <= Q(`s_1 * v_1 + ... + s_n * v_n`)
for all `s_i` where GCD(`s_k, ... s_n`) = 1.
Note: When Q has dim <= 4 we can take all `s_i` in {1, 0, -1}.
References:
Schulze-Pillot's paper on "An algorithm for computing genera
of ternary and quaternary quadratic forms", p138.
Donaldson's 1979 paper "Minkowski Reduction of Integral
Matrices", p203.
EXAMPLES::
sage: Q = QuadraticForm(ZZ,4,[30,17,11,12,29,25,62,64,25,110])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 12 ]
[ * 29 25 62 ]
[ * * 64 25 ]
[ * * * 110 ]
sage: Q.minkowski_reduction_for_4vars__SP()
(Quadratic form in 4 variables over Integer Ring with coefficients:
[ 29 -17 25 4 ]
[ * 30 -11 5 ]
[ * * 64 0 ]
[ * * * 77 ]
,
[ 0 1 0 0]
[ 1 0 0 -1]
[ 0 0 1 0]
[ 0 0 0 1])
"""
R = self.base_ring()
n = self.dim()
interior_reduced_flag = False
Q = deepcopy(self)
M = matrix(R, n, n)
for i in range(n):
M[i, i] = 1
## Only allow 4-variable forms
if n != 4:
raise TypeError, "Oops! The given quadratic form has " + str(n) + \
" != 4 variables. =|"
## Step 1: Begin the reduction
done_flag = False
while done_flag == False:
## Loop through possible shorter vectors
done_flag = True
#print " j_range = ", range(n-1, -1, -1)
for j in range(n - 1, -1, -1):
for a_first in mrange([2 for i in range(j)]):
y = [x - 1
for x in a_first] + [1] + [0 for k in range(n - 1 - j)]
e_j = [0 for k in range(n)]
e_j[j] = 1
#print "j = ", j
## Reduce if a shorter vector is found
#print "y = ", y, " e_j = ", e_j, "\n"
if Q(y) < Q(e_j):
## Further n=4 computations
B_y_vec = Q.matrix() * vector(ZZ, y)
## SP's B = our self.matrix()/2
## SP's A = coeff matrix of his B
## Here we compute the double of both and compare.
B_sum = sum([abs(B_y_vec[i]) for i in range(4) if i != j])
A_sum = sum([abs(Q[i, j]) for i in range(4) if i != j])
B_max = max([abs(B_y_vec[i]) for i in range(4) if i != j])
A_max = max([abs(Q[i, j]) for i in range(4) if i != j])
if (B_sum < A_sum) or ((B_sum == A_sum) and
(B_max < A_max)):
## Create the transformation matrix
M_new = matrix(R, n, n)
for k in range(n):
M_new[k, k] = 1
for k in range(n):
M_new[k, j] = y[k]
## Perform the reduction and restart the loop
#print "Q_before = ", Q
Q = Q(M_new)
M = M * M_new
done_flag = False
## DIAGNOSTIC
#print "Q(y) = ", Q(y)
#print "Q(e_j) = ", Q(e_j)
#print "M_new = ", M_new
#print "Q_after = ", Q
#print
if not done_flag:
break
if not done_flag:
break
## Step 2: Order A by certain criteria
for i in range(4):
for j in range(i + 1, 4):
## Condition (a)
if (Q[i, i] > Q[j, j]):
Q.swap_variables(i, j, in_place=True)
M_new = matrix(R, n, n)
M_new[i, j] = -1
M_new[j, i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r, r] = 0
else:
M_new[r, r] = 1
M = M * M_new
elif (Q[i, i] == Q[j, j]):
i_sum = sum([abs(Q[i, k]) for k in range(4) if k != i])
j_sum = sum([abs(Q[j, k]) for k in range(4) if k != j])
## Condition (b)
if (i_sum > j_sum):
Q.swap_variables(i, j, in_place=True)
M_new = matrix(R, n, n)
M_new[i, j] = -1
M_new[j, i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r, r] = 0
else:
M_new[r, r] = 1
M = M * M_new
elif (i_sum == j_sum):
for k in [
2, 1, 0
]: ## TO DO: These steps are a little redundant...
Q1 = Q.matrix()
c_flag = True
for l in range(k + 1, 4):
c_flag = c_flag and (abs(Q1[i, l]) == abs(Q1[j,
l]))
## Condition (c)
if c_flag and (abs(Q1[i, k]) > abs(Q1[j, k])):
Q.swap_variables(i, j, in_place=True)
M_new = matrix(R, n, n)
M_new[i, j] = -1
M_new[j, i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r, r] = 0
else:
M_new[r, r] = 1
M = M * M_new
## Step 3: Order the signs
for i in range(4):
if Q[i, 3] < 0:
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R, n, n)
for r in range(4):
if r == i:
M_new[r, r] = -1
else:
M_new[r, r] = 1
M = M * M_new
for i in range(4):
j = 3
while (Q[i, j] == 0):
j += -1
if (Q[i, j] < 0):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R, n, n)
for r in range(4):
if r == i:
M_new[r, r] = -1
else:
M_new[r, r] = 1
M = M * M_new
if Q[1, 2] < 0:
## Test a row 1 sign change
if (Q[1,3] <= 0 and \
((Q[1,3] < 0) or (Q[1,3] == 0 and Q[1,2] < 0) \
or (Q[1,3] == 0 and Q[1,2] == 0 and Q[1,1] < 0))):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R, n, n)
for r in range(4):
if r == i:
M_new[r, r] = -1
else:
M_new[r, r] = 1
M = M * M_new
elif (Q[2,3] <= 0 and \
((Q[2,3] < 0) or (Q[2,3] == 0 and Q[2,2] < 0) \
or (Q[2,3] == 0 and Q[2,2] == 0 and Q[2,1] < 0))):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R, n, n)
for r in range(4):
if r == i:
M_new[r, r] = -1
else:
M_new[r, r] = 1
M = M * M_new
## Return the results
return Q, M
def minkowski_reduction(self):
"""
Find a Minkowski-reduced form equivalent to the given one.
This means that
.. math::
Q(v_k) <= Q(s_1 * v_1 + ... + s_n * v_n)
for all `s_i` where GCD`(s_k, ... s_n) = 1`.
Note: When Q has dim <= 4 we can take all `s_i` in {1, 0, -1}.
References:
Schulze-Pillot's paper on "An algorithm for computing genera
of ternary and quaternary quadratic forms", p138.
Donaldson's 1979 paper "Minkowski Reduction of Integral
Matrices", p203.
EXAMPLES::
sage: Q = QuadraticForm(ZZ,4,[30,17,11,12,29,25,62,64,25,110])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 12 ]
[ * 29 25 62 ]
[ * * 64 25 ]
[ * * * 110 ]
sage: Q.minkowski_reduction()
(Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 -5 ]
[ * 29 25 4 ]
[ * * 64 0 ]
[ * * * 77 ]
,
[ 1 0 0 0]
[ 0 1 0 -1]
[ 0 0 1 0]
[ 0 0 0 1])
"""
R = self.base_ring()
n = self.dim()
interior_reduced_flag = False
Q = deepcopy(self)
M = matrix(R, n, n)
for i in range(n):
M[i, i] = 1
## Begin the reduction
done_flag = False
while done_flag == False:
## Loop through possible shorted vectors until
done_flag = True
#print " j_range = ", range(n-1, -1, -1)
for j in range(n - 1, -1, -1):
for a_first in mrange([2 for i in range(j)]):
y = [x - 1
for x in a_first] + [1] + [0 for k in range(n - 1 - j)]
e_j = [0 for k in range(n)]
e_j[j] = 1
#print "j = ", j
## Reduce if a shorter vector is found
#print "y = ", y, " e_j = ", e_j, "\n"
if Q(y) < Q(e_j):
## Create the transformation matrix
M_new = matrix(R, n, n)
for k in range(n):
M_new[k, k] = 1
for k in range(n):
M_new[k, j] = y[k]
## Perform the reduction and restart the loop
#print "Q_before = ", Q
Q = Q(M_new)
M = M * M_new
done_flag = False
## DIAGNOSTIC
#print "Q(y) = ", Q(y)
#print "Q(e_j) = ", Q(e_j)
#print "M_new = ", M_new
#print "Q_after = ", Q
#print
if not done_flag:
break
if not done_flag:
break
## Return the results
return Q, M
def Watson_mass_at_2(self):
"""
Returns the local mass of the quadratic form when `p=2`, according
to Watson's Theorem 1 of "The 2-adic density of a quadratic form"
in Mathematika 23 (1976), pp 94--106.
INPUT:
none
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.Watson_mass_at_2() ## WARNING: WE NEED TO CHECK THIS CAREFULLY!
384
"""
## Make a 0-dim'l quadratic form (for initialization purposes)
Null_Form = copy.deepcopy(self)
Null_Form.__init__(ZZ, 0)
## Step 0: Compute Jordan blocks and bounds of the scales to keep track of
Jordan_Blocks = self.jordan_blocks_by_scale_and_unimodular(2)
scale_list = [B[0] for B in Jordan_Blocks]
s_min = min(scale_list)
s_max = max(scale_list)
## Step 1: Compute dictionaries of the diagonal block and 2x2 block for each scale
diag_dict = dict((i, Null_Form) for i in range(s_min-2, s_max + 4)) ## Initialize with the zero form
dim2_dict = dict((i, Null_Form) for i in range(s_min, s_max + 4)) ## Initialize with the zero form
for (s,L) in Jordan_Blocks:
i = 0
while (i < L.dim()-1) and (L[i,i+1] == 0): ## Find where the 2x2 blocks start
i = i + 1
if i < (L.dim() - 1):
diag_dict[s] = L.extract_variables(range(i)) ## Diagonal Form
dim2_dict[s+1] = L.extract_variables(range(i, L.dim())) ## Non-diagonal Form
else:
diag_dict[s] = L
#print "diag_dict = ", diag_dict
#print "dim2_dict = ", dim2_dict
#print "Jordan_Blocks = ", Jordan_Blocks
## Step 2: Compute three dictionaries of invariants (for n_j, m_j, nu_j)
n_dict = dict((j,0) for j in range(s_min+1, s_max+2))
m_dict = dict((j,0) for j in range(s_min, s_max+4))
for (s,L) in Jordan_Blocks:
n_dict[s+1] = L.dim()
if diag_dict[s].dim() == 0:
m_dict[s+1] = ZZ(1)/ZZ(2) * L.dim()
else:
m_dict[s+1] = floor(ZZ(L.dim() - 1) / ZZ(2))
#print " ==>", ZZ(L.dim() - 1) / ZZ(2), floor(ZZ(L.dim() - 1) / ZZ(2))
nu_dict = dict((j,n_dict[j+1] - 2*m_dict[j+1]) for j in range(s_min, s_max+1))
nu_dict[s_max+1] = 0
#print "n_dict = ", n_dict
#print "m_dict = ", m_dict
#print "nu_dict = ", nu_dict
## Step 3: Compute the e_j dictionary
eps_dict = {}
for j in range(s_min, s_max+3):
two_form = (diag_dict[j-2] + diag_dict[j] + dim2_dict[j]).scale_by_factor(2)
j_form = (two_form + diag_dict[j-1]).base_change_to(IntegerModRing(4))
if j_form.dim() == 0:
eps_dict[j] = 1
else:
iter_vec = [4] * j_form.dim()
alpha = sum([True for x in mrange(iter_vec) if j_form(x) == 0])
beta = sum([True for x in mrange(iter_vec) if j_form(x) == 2])
if alpha > beta:
eps_dict[j] = 1
elif alpha == beta:
eps_dict[j] = 0
else:
eps_dict[j] = -1
#print "eps_dict = ", eps_dict
## Step 4: Compute the quantities nu, q, P, E for the local mass at 2
nu = sum([j * n_dict[j] * (ZZ(n_dict[j] + 1) / ZZ(2) + \
sum([n_dict[r] for r in range(j+1, s_max+2)])) for j in range(s_min+1, s_max+2)])
q = sum([sgn(nu_dict[j-1] * (n_dict[j] + sgn(nu_dict[j]))) for j in range(s_min+1, s_max+2)])
P = prod([ prod([1 - QQ(4)**(-i) for i in range(1, m_dict[j]+1)]) for j in range(s_min+1, s_max+2)])
E = prod([ZZ(1)/ZZ(2) * (1 + eps_dict[j] * QQ(2)**(-m_dict[j])) for j in range(s_min, s_max+3)])
#print "\nFinal Summary:"
#print "nu =", nu
#print "q = ", q
#print "P = ", P
#print "E = ", E
## Step 5: Compute the local mass for the prime 2.
mass_at_2 = QQ(2)**(nu - q) * P / E
return mass_at_2
def minkowski_reduction_for_4vars__SP(self):
"""
Find a Minkowski-reduced form equivalent to the given one.
This means that
Q(`v_k`) <= Q(`s_1 * v_1 + ... + s_n * v_n`)
for all `s_i` where GCD(`s_k, ... s_n`) = 1.
Note: When Q has dim <= 4 we can take all `s_i` in {1, 0, -1}.
References:
Schulze-Pillot's paper on "An algorithm for computing genera
of ternary and quaternary quadratic forms", p138.
Donaldson's 1979 paper "Minkowski Reduction of Integral
Matrices", p203.
EXAMPLES::
sage: Q = QuadraticForm(ZZ,4,[30,17,11,12,29,25,62,64,25,110])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 12 ]
[ * 29 25 62 ]
[ * * 64 25 ]
[ * * * 110 ]
sage: Q.minkowski_reduction_for_4vars__SP()
(
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 29 -17 25 4 ]
[ * 30 -11 5 ]
[ * * 64 0 ]
[ * * * 77 ] ,
<BLANKLINE>
[ 0 1 0 0]
[ 1 0 0 -1]
[ 0 0 1 0]
[ 0 0 0 1]
)
"""
R = self.base_ring()
n = self.dim()
interior_reduced_flag = False
Q = deepcopy(self)
M = matrix(R, n, n)
for i in range(n):
M[i,i] = 1
## Only allow 4-variable forms
if n != 4:
raise TypeError("Oops! The given quadratic form has " + str(n) + \
" != 4 variables. =|")
## Step 1: Begin the reduction
done_flag = False
while not done_flag:
## Loop through possible shorter vectors
done_flag = True
#print " j_range = ", range(n-1, -1, -1)
for j in range(n-1, -1, -1):
for a_first in mrange([2 for i in range(j)]):
y = [x-1 for x in a_first] + [1] + [0 for k in range(n-1-j)]
e_j = [0 for k in range(n)]
e_j[j] = 1
#print "j = ", j
## Reduce if a shorter vector is found
#print "y = ", y, " e_j = ", e_j, "\n"
if Q(y) < Q(e_j):
## Further n=4 computations
B_y_vec = Q.matrix() * vector(ZZ, y)
## SP's B = our self.matrix()/2
## SP's A = coeff matrix of his B
## Here we compute the double of both and compare.
B_sum = sum([abs(B_y_vec[i]) for i in range(4) if i != j])
A_sum = sum([abs(Q[i,j]) for i in range(4) if i != j])
B_max = max([abs(B_y_vec[i]) for i in range(4) if i != j])
A_max = max([abs(Q[i,j]) for i in range(4) if i != j])
if (B_sum < A_sum) or ((B_sum == A_sum) and (B_max < A_max)):
## Create the transformation matrix
M_new = matrix(R, n, n)
for k in range(n):
M_new[k,k] = 1
for k in range(n):
M_new[k,j] = y[k]
## Perform the reduction and restart the loop
#print "Q_before = ", Q
Q = Q(M_new)
M = M * M_new
done_flag = False
## DIAGNOSTIC
#print "Q(y) = ", Q(y)
#print "Q(e_j) = ", Q(e_j)
#print "M_new = ", M_new
#print "Q_after = ", Q
#print
if not done_flag:
break
if not done_flag:
break
## Step 2: Order A by certain criteria
for i in range(4):
for j in range(i+1,4):
## Condition (a)
if (Q[i,i] > Q[j,j]):
Q.swap_variables(i,j,in_place=True)
M_new = matrix(R,n,n)
M_new[i,j] = -1
M_new[j,i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r,r] = 0
else:
M_new[r,r] = 1
M = M * M_new
elif (Q[i,i] == Q[j,j]):
i_sum = sum([abs(Q[i,k]) for k in range(4) if k != i])
j_sum = sum([abs(Q[j,k]) for k in range(4) if k != j])
## Condition (b)
if (i_sum > j_sum):
Q.swap_variables(i,j,in_place=True)
M_new = matrix(R,n,n)
M_new[i,j] = -1
M_new[j,i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r,r] = 0
else:
M_new[r,r] = 1
M = M * M_new
elif (i_sum == j_sum):
for k in [2,1,0]: ## TO DO: These steps are a little redundant...
Q1 = Q.matrix()
c_flag = True
for l in range(k+1,4):
c_flag = c_flag and (abs(Q1[i,l]) == abs(Q1[j,l]))
## Condition (c)
if c_flag and (abs(Q1[i,k]) > abs(Q1[j,k])):
Q.swap_variables(i,j,in_place=True)
M_new = matrix(R,n,n)
M_new[i,j] = -1
M_new[j,i] = 1
for r in range(4):
if (r == i) or (r == j):
M_new[r,r] = 0
else:
M_new[r,r] = 1
M = M * M_new
## Step 3: Order the signs
for i in range(4):
if Q[i,3] < 0:
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R,n,n)
for r in range(4):
if r == i:
M_new[r,r] = -1
else:
M_new[r,r] = 1
M = M * M_new
for i in range(4):
j = 3
while (Q[i,j] == 0):
j += -1
if (Q[i,j] < 0):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R,n,n)
for r in range(4):
if r == i:
M_new[r,r] = -1
else:
M_new[r,r] = 1
M = M * M_new
if Q[1,2] < 0:
## Test a row 1 sign change
if (Q[1,3] <= 0 and \
((Q[1,3] < 0) or (Q[1,3] == 0 and Q[1,2] < 0) \
or (Q[1,3] == 0 and Q[1,2] == 0 and Q[1,1] < 0))):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R,n,n)
for r in range(4):
if r == i:
M_new[r,r] = -1
else:
M_new[r,r] = 1
M = M * M_new
elif (Q[2,3] <= 0 and \
((Q[2,3] < 0) or (Q[2,3] == 0 and Q[2,2] < 0) \
or (Q[2,3] == 0 and Q[2,2] == 0 and Q[2,1] < 0))):
Q.multiply_variable(-1, i, in_place=True)
M_new = matrix(R,n,n)
for r in range(4):
if r == i:
M_new[r,r] = -1
else:
M_new[r,r] = 1
M = M * M_new
## Return the results
return Q, M
def minkowski_reduction(self):
"""
Find a Minkowski-reduced form equivalent to the given one.
This means that
.. MATH::
Q(v_k) <= Q(s_1 * v_1 + ... + s_n * v_n)
for all `s_i` where GCD`(s_k, ... s_n) = 1`.
Note: When Q has dim <= 4 we can take all `s_i` in {1, 0, -1}.
References:
Schulze-Pillot's paper on "An algorithm for computing genera
of ternary and quaternary quadratic forms", p138.
Donaldson's 1979 paper "Minkowski Reduction of Integral
Matrices", p203.
EXAMPLES::
sage: Q = QuadraticForm(ZZ,4,[30,17,11,12,29,25,62,64,25,110])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 12 ]
[ * 29 25 62 ]
[ * * 64 25 ]
[ * * * 110 ]
sage: Q.minkowski_reduction()
(
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 -5 ]
[ * 29 25 4 ]
[ * * 64 0 ]
[ * * * 77 ] ,
<BLANKLINE>
[ 1 0 0 0]
[ 0 1 0 -1]
[ 0 0 1 0]
[ 0 0 0 1]
)
"""
R = self.base_ring()
n = self.dim()
interior_reduced_flag = False
Q = deepcopy(self)
M = matrix(R, n, n)
for i in range(n):
M[i,i] = 1
## Begin the reduction
done_flag = False
while not done_flag:
## Loop through possible shorted vectors until
done_flag = True
#print " j_range = ", range(n-1, -1, -1)
for j in range(n-1, -1, -1):
for a_first in mrange([2 for i in range(j)]):
y = [x-1 for x in a_first] + [1] + [0 for k in range(n-1-j)]
e_j = [0 for k in range(n)]
e_j[j] = 1
#print "j = ", j
## Reduce if a shorter vector is found
#print "y = ", y, " e_j = ", e_j, "\n"
if Q(y) < Q(e_j):
## Create the transformation matrix
M_new = matrix(R, n, n)
for k in range(n):
M_new[k,k] = 1
for k in range(n):
M_new[k,j] = y[k]
## Perform the reduction and restart the loop
#print "Q_before = ", Q
Q = Q(M_new)
M = M * M_new
done_flag = False
## DIAGNOSTIC
#print "Q(y) = ", Q(y)
#print "Q(e_j) = ", Q(e_j)
#print "M_new = ", M_new
#print "Q_after = ", Q
#print
if not done_flag:
break
if not done_flag:
break
## Return the results
return Q, M
def is_globally_equivalent_to(self, other, return_matrix=False, check_theta_to_precision='sturm', check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError, "not a definite form in QuadraticForm.is_globally_equivalent_to()"
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError, "Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(self, return_matrix) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision != None:
if self.theta_series(check_theta_to_precision, var_str='', safe_flag=False) != other.theta_series(check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([short_vectors_of_other[self_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
def minkowski_reduction(self):
"""
Find a Minkowski-reduced form equivalent to the given one.
This means that
.. MATH::
Q(v_k) <= Q(s_1 * v_1 + ... + s_n * v_n)
for all `s_i` where GCD`(s_k, ... s_n) = 1`.
Note: When Q has dim <= 4 we can take all `s_i` in {1, 0, -1}.
References:
Schulze-Pillot's paper on "An algorithm for computing genera
of ternary and quaternary quadratic forms", p138.
Donaldson's 1979 paper "Minkowski Reduction of Integral
Matrices", p203.
EXAMPLES::
sage: Q = QuadraticForm(ZZ,4,[30, 17, 11, 12, 29, 25, 62, 64, 25, 110])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 12 ]
[ * 29 25 62 ]
[ * * 64 25 ]
[ * * * 110 ]
sage: Q.minkowski_reduction()
(
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 30 17 11 -5 ]
[ * 29 25 4 ]
[ * * 64 0 ]
[ * * * 77 ] ,
<BLANKLINE>
[ 1 0 0 0]
[ 0 1 0 -1]
[ 0 0 1 0]
[ 0 0 0 1]
)
::
sage: Q=QuadraticForm(ZZ,4,[1, -2, 0, 0, 2, 0, 0, 2, 0, 2])
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 1 -2 0 0 ]
[ * 2 0 0 ]
[ * * 2 0 ]
[ * * * 2 ]
sage: Q.minkowski_reduction()
(
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 1 0 0 0 ]
[ * 1 0 0 ]
[ * * 2 0 ]
[ * * * 2 ] ,
<BLANKLINE>
[1 1 0 0]
[0 1 0 0]
[0 0 1 0]
[0 0 0 1]
)
::
sage: Q=QuadraticForm(ZZ,5,[2,2,0,0,0,2,2,0,0,2,2,0,2,2,2])
sage: Q.Gram_matrix()
[2 1 0 0 0]
[1 2 1 0 0]
[0 1 2 1 0]
[0 0 1 2 1]
[0 0 0 1 2]
sage: Q.minkowski_reduction()
Traceback (most recent call last):
...
NotImplementedError: This algorithm is only for dimensions less than 5
"""
from sage.quadratic_forms.quadratic_form import QuadraticForm
from sage.quadratic_forms.quadratic_form import matrix
if not self.is_positive_definite():
raise TypeError(
"Minkowksi reduction only works for positive definite forms")
if self.dim() > 4:
raise NotImplementedError(
"This algorithm is only for dimensions less than 5")
R = self.base_ring()
n = self.dim()
Q = deepcopy(self)
M = matrix(R, n, n)
for i in range(n):
M[i, i] = 1
## Begin the reduction
done_flag = False
while not done_flag:
## Loop through possible shorted vectors until
done_flag = True
#print " j_range = ", range(n-1, -1, -1)
for j in range(n - 1, -1, -1):
for a_first in mrange([3 for i in range(j)]):
y = [x - 1
for x in a_first] + [1] + [0 for k in range(n - 1 - j)]
e_j = [0 for k in range(n)]
e_j[j] = 1
#print "j = ", j
## Reduce if a shorter vector is found
#print "y = ", y, " e_j = ", e_j, "\n"
if Q(y) < Q(e_j):
## Create the transformation matrix
M_new = matrix(R, n, n)
for k in range(n):
M_new[k, k] = 1
for k in range(n):
M_new[k, j] = y[k]
## Perform the reduction and restart the loop
#print "Q_before = ", Q
Q = QuadraticForm(M_new.transpose() * Q.matrix() * M_new)
M = M * M_new
done_flag = False
## DIAGNOSTIC
#print "Q(y) = ", Q(y)
#print "Q(e_j) = ", Q(e_j)
#print "M_new = ", M_new
#print "Q_after = ", Q
#print
if not done_flag:
break
if not done_flag:
break
## Return the results
return Q, M
def automorphisms(self):
"""
Return a list of the automorphisms of the quadratic form.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.number_of_automorphisms() # optional -- souvigner
48
sage: 2^3 * factorial(3)
48
sage: len(Q.automorphisms())
48
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,5,7])
sage: Q.number_of_automorphisms() # optional -- souvigner
16
sage: aut = Q.automorphisms()
sage: len(aut)
16
sage: print([Q(M) == Q for M in aut])
[True, True, True, True, True, True, True, True, True, True, True, True, True, True, True, True]
sage: Q = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q.automorphisms()
[
[1 0 0] [-1 0 0]
[0 1 0] [ 0 -1 0]
[0 0 1], [ 0 0 -1]
]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.automorphisms()
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.automorphisms()
"""
## only for definite forms
if not self.is_definite():
raise ValueError("not a definite form in QuadraticForm.automorphisms()")
## Check for a cached value
try:
return self.__automorphisms
except AttributeError:
pass
## Find a basis of short vectors, and their lengths
basis, pivot_lengths = self.basis_of_short_vectors(show_lengths=True)
## List the relevant vectors by length
max_len = max(pivot_lengths)
vector_list_by_length = self.short_primitive_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis).transpose()
Ainv = A.inverse()
# A1 = A.inverse() * A.det()
# Q1 = A1.transpose() * self.matrix() * A1 ## This is the matrix of Q
# Q = self.matrix() * A.det()**2
Q2 = A.transpose() * self.matrix() * A ## This is the matrix of Q in the new basis
Q3 = self.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
# ct = 0
## DIAGNOSTIC
# print "n = " + str(n)
# print "pivot_lengths = " + str(pivot_lengths)
# print "vector_list_by_length = " + str(vector_list_by_length)
# print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange([len(vector_list_by_length[pivot_lengths[i]]) for i in range(n)]):
M = Matrix([vector_list_by_length[pivot_lengths[i]][index_vec[i]] for i in range(n)]).transpose()
# Q1 = self.matrix()
# if self(M) == self:
# ct += 1
# print "ct = ", ct, " M = "
# print M
# print
if M.transpose() * Q3 * M == Q2: ## THIS DOES THE SAME THING! =(
Auto_list.append(M * Ainv)
## Cache the answer and return the list
self.__automorphisms = Auto_list
self.__number_of_automorphisms = len(Auto_list)
return Auto_list
def is_globally_equivalent_to(self,
other,
return_matrix=False,
check_theta_to_precision='sturm',
check_local_equivalence=True):
"""
Determines if the current quadratic form is equivalent to the
given form over ZZ. If return_matrix is True, then we also return
the transformation matrix M so that self(M) == other.
INPUT:
a QuadraticForm
OUTPUT:
boolean, and optionally a matrix
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: M = Matrix(ZZ, 4, 4, [1,2,0,0, 0,1,0,0, 0,0,1,0, 0,0,0,1])
sage: Q1 = Q(M)
sage: Q.(Q1) # optional -- souvigner
True
sage: MM = Q.is_globally_equivalent_to(Q1, return_matrix=True) # optional -- souvigner
sage: Q(MM) == Q1 # optional -- souvigner
True
::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q3 = QuadraticForm(ZZ, 3, [8, 6, 5, 3, 4, 2])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_globally_equivalent_to(Q3) # optional -- souvigner
True
sage: M = Q1.is_globally_equivalent_to(Q3, True) ; M # optional -- souvigner
[-1 -1 0]
[ 1 1 1]
[-1 0 0]
sage: Q1(M) == Q3 # optional -- souvigner
True
::
sage: Q = DiagonalQuadraticForm(ZZ, [1, -1])
sage: Q.is_globally_equivalent_to(Q)
Traceback (most recent call last):
...
ValueError: not a definite form in QuadraticForm.is_globally_equivalent_to()
"""
## only for definite forms
if not self.is_definite():
raise ValueError(
"not a definite form in QuadraticForm.is_globally_equivalent_to()")
## Check that other is a QuadraticForm
#if not isinstance(other, QuadraticForm):
if not is_QuadraticForm(other):
raise TypeError(
"Oops! You must compare two quadratic forms, but the argument is not a quadratic form. =("
)
## Now use the Souvigner code by default! =)
return other.is_globally_equivalent__souvigner(
self, return_matrix
) ## Note: We switch this because the Souvigner code has the opposite mapping convention to us. (It takes the second argument to the first!)
## ---------------------------------- Unused Code below ---------------------------------------------------------
## Check if the forms are locally equivalent
if (check_local_equivalence == True):
if not self.is_locally_equivalent_to(other):
return False
## Check that the forms have the same theta function up to the desired precision (this can be set so that it determines the cusp form)
if check_theta_to_precision is not None:
if self.theta_series(
check_theta_to_precision, var_str='',
safe_flag=False) != other.theta_series(
check_theta_to_precision, var_str='', safe_flag=False):
return False
## Make all possible matrices which give an isomorphism -- can we do this more intelligently?
## ------------------------------------------------------------------------------------------
## Find a basis of short vectors for one form, and try to match them with vectors of that length in the other one.
basis_for_self, self_lengths = self.basis_of_short_vectors(
show_lengths=True)
max_len = max(self_lengths)
short_vectors_of_other = other.short_vector_list_up_to_length(max_len + 1)
## Make the matrix A:e_i |--> v_i to our new basis.
A = Matrix(basis_for_self).transpose()
Q2 = A.transpose() * self.matrix(
) * A ## This is the matrix of 'self' in the new basis
Q3 = other.matrix()
## Determine all automorphisms
n = self.dim()
Auto_list = []
## DIAGNOSTIC
#print "n = " + str(n)
#print "pivot_lengths = " + str(pivot_lengths)
#print "vector_list_by_length = " + str(vector_list_by_length)
#print "length of vector_list_by_length = " + str(len(vector_list_by_length))
for index_vec in mrange(
[len(short_vectors_of_other[self_lengths[i]]) for i in range(n)]):
M = Matrix([
short_vectors_of_other[self_lengths[i]][index_vec[i]]
for i in range(n)
]).transpose()
if M.transpose() * Q3 * M == Q2:
if return_matrix:
return A * M.inverse()
else:
return True
## If we got here, then there is no isomorphism
return False
