Python Matrix.dimension方法代码示例

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import dimension [as 别名]
    def get_cocycle_from_elliptic_curve(self,E,sign = 1,use_magma = True):
        if sign == 0:
            return self.get_cocycle_from_elliptic_curve(E,1,use_magma) + self.get_cocycle_from_elliptic_curve(E,-1,use_magma)
        if not sign in [1, -1]:
            raise NotImplementedError
        F = self.group().base_ring()
        if F.signature()[1] == 0 or (F.signature() == (0,1) and 'G' not in self.group()._grouptype):
            K = (self.hecke_matrix(oo).transpose()-sign).kernel().change_ring(QQ)
        else:
            K = Matrix(QQ,self.dimension(),self.dimension(),0).kernel()
        disc = self.S_arithgroup().Gpn._O_discriminant
        discnorm = disc.norm()
        try:
            N = ZZ(discnorm.gen())
        except AttributeError:
            N = ZZ(discnorm)

        if F == QQ:
            x = QQ['x'].gen()
            F = NumberField(x,names='a')
            E = E.change_ring(F)

        def getap(q):
            if F == QQ:
                return E.ap(q)
            else:
                Q = F.ideal(q).factor()[0][0]
                return ZZ(Q.norm() + 1 - E.reduction(Q).count_points())

        q = ZZ(1)
        g0 = None
        while K.dimension() > 1:
            q = q.next_prime()
            for qq,e in F.ideal(q).factor():
                if  ZZ(qq.norm()).is_prime() and not qq.divides(F.ideal(disc.gens_reduced()[0])):
                    try:
                        ap = getap(qq)
                    except (ValueError,ArithmeticError):
                        continue
                    try:
                        K1 = (self.hecke_matrix(qq.gens_reduced()[0],g0 = g0,use_magma = use_magma).transpose()-ap).kernel()
                    except RuntimeError:
                        continue
                    K = K.intersection(K1)
        if K.dimension() != 1:
            raise ValueError,'Did not obtain a one-dimensional space corresponding to E'
        col = [ZZ(o) for o in (K.denominator()*K.matrix()).list()]
        return sum([a * self.gen(i) for i,a in enumerate(col) if a != 0],self(0))

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import dimension [as 别名]
    def get_rational_cocycle_from_ap(self,getap,sign = 1,use_magma = True):
        F = self.group().base_ring()
        if F.signature()[1] == 0 or (F.signature() == (0,1) and 'G' not in self.group()._grouptype):
            K = (self.hecke_matrix(oo).transpose()-sign).kernel().change_ring(QQ)
        else:
            K = Matrix(QQ,self.dimension(),self.dimension(),0).kernel()

        disc = self.S_arithgroup().Gpn._O_discriminant
        discnorm = disc.norm()
        try:
            N = ZZ(discnorm.gen())
        except AttributeError:
            N = ZZ(discnorm)

        if F == QQ:
            x = QQ['x'].gen()
            F = NumberField(x,names='a')
        q = ZZ(1)
        g0 = None
        while K.dimension() > 1:
            q = q.next_prime()
            for qq,e in F.ideal(q).factor():
                if  ZZ(qq.norm()).is_prime() and not qq.divides(F.ideal(disc.gens_reduced()[0])):
                    try:
                        ap = getap(qq)
                    except (ValueError,ArithmeticError):
                        continue
                    try:
                        K1 = (self.hecke_matrix(qq.gens_reduced()[0],g0 = g0,use_magma = use_magma).transpose()-ap).kernel()
                    except RuntimeError:
                        continue
                    K = K.intersection(K1)
        if K.dimension() != 1:
            raise ValueError,'Group does not have the required system of eigenvalues'

        col = [ZZ(o) for o in (K.denominator()*K.matrix()).list()]
        return sum([ a * self.gen(i) for i,a in enumerate(col) if a != 0], self(0))

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import dimension [as 别名]
    def get_twodim_cocycle(self,sign = 1,use_magma = True,bound = 5, hecke_data = None, return_all = False, outfile = None):
        F = self.group().base_ring()
        if F == QQ:
            F = NumberField(PolynomialRing(QQ,'x').gen(),names='r')
        component_list = []
        good_components = []
        if F.signature()[1] == 0 or (F.signature() == (0,1) and 'G' not in self.group()._grouptype):
            Tinf = self.hecke_matrix(oo).transpose()
            K = (Tinf-sign).kernel().change_ring(QQ)
            if K.dimension() >= 2:
                component_list.append((K, [(oo,Tinf)]))
            fwrite('Too charpoly = %s'%Tinf.charpoly().factor(),outfile)
        else:
            K = Matrix(QQ,self.dimension(),self.dimension(),0).kernel()
            if K.dimension() >= 2:
                component_list.append((K, []))
        disc = self.S_arithgroup().Gpn._O_discriminant
        discnorm = disc.norm()
        try:
            N = ZZ(discnorm.gen())
        except AttributeError:
            N = ZZ(discnorm)

        if F == QQ:
            x = QQ['x'].gen()
            F = NumberField(x,names='a')
        q = ZZ(1)
        g0 = None
        num_hecke_operators = 0
        if hecke_data is not None:
            qq = F.ideal(hecke_data[0])
            pol = hecke_data[1]
            Aq = self.hecke_matrix(qq.gens_reduced()[0], use_magma = use_magma).transpose().change_ring(QQ)
            fwrite('ell = (%s,%s), T_ell charpoly = %s'%(qq.norm(), qq.gens_reduced()[0], Aq.charpoly().factor()),outfile)
            U0 = component_list[0][0].intersection(pol.subs(Aq).left_kernel())
            if U0.dimension() != 2:
                raise ValueError('Hecke data does not suffice to cut out space')
            good_component = (U0.denominator() * U0,component_list[0][1] + [(qq.gens_reduced()[0],Aq)])
            row0 = good_component[0].matrix().rows()[0]
            col0 = [QQ(o) for o in row0.list()]
            clcm = lcm([o.denominator() for o in col0])
            col0 = [ZZ(clcm * o ) for o in col0]
            # phi1 = sum([a * phi for a,phi in zip(col0,self.gens())],self(0))
            # phi2 = self.apply_hecke_operator(phi1,qq.gens_reduced()[0], use_magma = use_magma)
            # return [phi1, phi2], [(ell, o.restrict(good_component[0])) for ell, o in good_component[1]]
            flist = []
            for row0 in good_component[0].matrix().rows():
                col0 = [QQ(o) for o in row0.list()]
                clcm = lcm([o.denominator() for o in col0])
                col0 = [ZZ(clcm * o ) for o in col0]
                flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0)))
            return flist,[(ell, o.restrict(good_component[0])) for ell, o in good_component[1]]
        while len(component_list) > 0 and num_hecke_operators < bound:
            verbose('num_hecke_ops = %s'%num_hecke_operators)
            verbose('len(components_list) = %s'%len(component_list))
            q = q.next_prime()
            verbose('q = %s'%q)
            fact = F.ideal(q).factor()
            dfact = F.ideal(disc.gens_reduced()[0]).factor()
            for qq,e in fact:
                verbose('Trying qq = %s'%qq)
                if qq in [o for o,_ in dfact]:
                    verbose('Skipping because qq divides D...')
                    continue
                if  ZZ(qq.norm()).is_prime() and not qq.divides(F.ideal(disc.gens_reduced()[0])):
                    try:
                        Aq = self.hecke_matrix(qq.gens_reduced()[0],g0 = g0,use_magma = use_magma).transpose().change_ring(QQ)
                    except (RuntimeError,TypeError) as e:
                        verbose('Skipping qq (=%s) because Hecke matrix could not be computed...'%qq.gens_reduced()[0])
                        continue
                    except KeyboardInterrupt:
                        verbose('Skipping qq (=%s) by user request...'%qq.gens_reduced()[0])
                        num_hecke_operators += 1
                        sleep(1)
                        continue
                    verbose('Computed hecke matrix at qq = %s'%qq)
                    fwrite('ell = (%s,%s), T_ell charpoly = %s'%(qq.norm(), qq.gens_reduced()[0], Aq.charpoly().factor()),outfile)
                    old_component_list = component_list
                    component_list = []
                    num_hecke_operators += 1
                    for U,hecke_data in old_component_list:
                        V = Aq.decomposition_of_subspace(U)
                        for U0,is_irred in V:
                            if U0.dimension() == 1:
                                continue
                            if U0.dimension() == 2 and is_irred:
                                good_components.append((U0.denominator() * U0,hecke_data+[(qq.gens_reduced()[0],Aq)]))
                            else: # U0.dimension() > 2 or not is_irred
                                component_list.append((U0.denominator() * U0,hecke_data + [(qq.gens_reduced()[0],Aq)]))
                    if len(good_components) > 0 and not return_all:
                        flist = []
                        for row0 in good_components[0][0].matrix().rows():
                            col0 = [QQ(o) for o in row0.list()]
                            clcm = lcm([o.denominator() for o in col0])
                            col0 = [ZZ(clcm * o ) for o in col0]
                            flist.append(sum([a * phi for a,phi in zip(col0,self.gens())],self(0)))
                        return flist,[(ell, o.restrict(good_components[0][0])) for ell, o in good_components[0][1]]
                    if len(component_list) == 0 or num_hecke_operators >= bound:
                        break

#.........这里部分代码省略.........

# 需要导入模块: from sage.matrix.constructor import Matrix [as 别名]
# 或者: from sage.matrix.constructor.Matrix import dimension [as 别名]
    def get_rational_cocycle(self,sign = 1,use_magma = True,bound = 3, return_all = False):
        F = self.group().base_ring()
        if F.signature()[1] == 0 or (F.signature()[1] == 1 and 'G' not in self.group()._grouptype):
            K = (self.hecke_matrix(oo).transpose()-sign).kernel().change_ring(QQ)
        else:
            K = Matrix(QQ,self.dimension(),self.dimension(),0).kernel()

        component_list = []
        good_components = []
        if K.dimension() == 1:
            good_components.append(K)
        else:
            component_list.append(K)

        disc = self.S_arithgroup().Gpn._O_discriminant
        discnorm = disc.norm()
        try:
            N = ZZ(discnorm.gen())
        except AttributeError:
            N = ZZ(discnorm)

        if F == QQ:
            x = QQ['x'].gen()
            F = NumberField(x,names='a')
        q = ZZ(1)
        g0 = None
        num_hecke_operators = 0
        while len(component_list) > 0 and num_hecke_operators < bound:
            verbose('num_hecke_ops = %s'%num_hecke_operators)
            verbose('len(components_list) = %s'%len(component_list))
            q = q.next_prime()
            for qq,e in F.ideal(q).factor():
                if  ZZ(qq.norm()).is_prime() and not qq.divides(F.ideal(disc.gens_reduced()[0])):
                    try:
                        Aq = self.hecke_matrix(qq.gens_reduced()[0],g0 = g0,use_magma = use_magma).transpose().change_ring(QQ)
                    except (RuntimeError,TypeError) as e:
                        continue
                    verbose('Computed hecke matrix at qq = %s'%qq)
                    old_component_list = component_list
                    component_list = []
                    num_hecke_operators += 1
                    for U in old_component_list:
                        V = Aq.decomposition_of_subspace(U)
                        for U0,is_irred in V:
                            if Aq.restrict(U0).eigenvalues()[0] == ZZ(qq.norm()) + 1:
                                continue # Do not take Eisenstein classes.
                            if U0.dimension() == 1:
                                good_components.append(U0)
                            elif is_irred:
                                # Bad
                                continue
                            else: # U0.dimension() > 1 and not is_irred
                                component_list.append(U0)
                    if len(good_components) > 0 and not return_all:
                        K = good_components[0]
                        col = [ZZ(o) for o in (K.denominator()*K.matrix()).list()]
                        return sum([a*self.gen(i) for i,a in enumerate(col) if a != 0],self(0))
                    if len(component_list) == 0 or num_hecke_operators >= bound:
                        break

        if len(good_components) == 0:
            raise ValueError('Group does not seem to be attached to an elliptic curve')
        else:
            if return_all:
                ans = []
                for K in good_components:
                    col = [ZZ(o) for o in (K.denominator()*K.matrix()).list()]
                    ans.append( sum([a*self.gen(i) for i,a in enumerate(col) if a != 0],self(0)))
                return ans
            else:
                K = good_components[0]
                col = [ZZ(o) for o in (K.denominator()*K.matrix()).list()]
                return sum([ a * self.gen(i) for i,a in enumerate(col) if a != 0], self(0))