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IsogenousIdealstH(���������I(2�Returns the left ideal of A generated by p^n and (a + b*x1)*x2^n, where Q = <a,b> with GCD(a,b) = 1. Here (q = p^n, x2^n) is the nth power of a split prime (p,x2) in Z[x2]. � J(������������0�|�$��#K(  __sig__31�$�$L( |�,������&M( IsogenousIdeal�'N(��������(�O(MReturns the level of a positive definite quadratic form having Gram matrix M.�/P( ��������H�|�<��2Q(  __sig__32P4h4R( |�D������5S(  FormLevel�7t8T(��������@�U(@Returns the number of classes in the Brandt groupoid of level N.<�<�<V(��������p��|�T��@W(  __sig__33HB�BX( |�\������Dv(  �����������ԡ�y(  CompareForms p z(��������ء{(2Returns the InnerProduct(x*M,y) with respect to M.L d |(99 ����������������}( __sig__5��~( �������( VReturns the matrix N such that M*N = N*M is the determinant times the identity matrix. �( �������������� �( __sig__6!" "�( �������|#Y( LevelClassNumberH�HIZ(��������X�[(DReturns the number of classes in the Brandt groupoid of level (N,M).PPP$$��������x�|�l� T](  __sig__34\U�U^( |�t������W_(*/home/was/magma/Kohel/quadratic_modules.mg� ���(*/home/was/magma/Kohel/quadratic_modules.mg� ���a( YReturns the unique Minkowski reduced representative for QF, and the transforming matrix. �a�ab( ��������������8gc( __sig__0h�h id( ���������je( MinkowskiReduction�l�lf(����������g( Returns the matrix obtained by a basis permutation such that QF[i,i] le QF[j,j] for all i le j. w�w�wh( ���������������zi( __sig__1{�{�{j( ��������<}�( ClassicalAdjoint� �(����������( ]Returns the determinant of the matrix obtained by removing the ith row and jth column from M. �( ���������� ������( __sig__7���( �� �����t�( MinorDeterminant���(����������(/Returns 1 if x and y are equal and 0 otherwise. 0 H �(����������8���,��!�( __sig__8"�"�"�( ��4�����k(  NormEchelon�~�~l(����������m(,�Returns true if and only if QF[i,i] le QF[j,j] and 2*|QF[i,j]| le M[i,i] for all i le j. This is weaker than the matrix being Minkowski reduced. �8�n( ���������������o( __sig__2����p( ����������q(>�Compare two matrices under the ordering obtained by defining as smaller the matrix with the smallest first diagonal element, or, in absolute value, the largest off-diagonal element, under an ordering of the entries. ��\�r(  ��������С��ġ8�s( __sig__3�����t( ��̡�����u( WReturn 1 if M1 is less than M2, 0 if M1 and M2 are equal, and -1 if M2 is less than M1.X�p���w( __sig__4����x(��ܡ����L�d����( KroneckerDelta �(��������0��(%/home/was/magma/Kohel/power_series.mg�XU��r_���d5���d5���d5���d5���d5��d5��d5��d5 �H��6 ��d5��d5(��d5,��d5\��d5|��d5���d5���d5���d5��d5��d5�� d5�� d5��d5��d5 �d5�d5(� d5,� d50�d54�'d58� �5h�7 z2�����5����o6���o6 ��o6� � �o6 �*�t6x �S�Q7� �E�^30 �~ z2� �� z2� �Z z2� �y z2 ���638�T z2p�b z2��B z2��l z2H�p z2h�u z2��g z2��= z2��P z2�If�30�1f�3H��-Z6��#�30�Ef�3P���'7��<R'4�����4�����6�p�*1 �9'w74�d��3l� &�5��8{1��1�����6D��ow5��<�X5���Em3��/�Q3�u2T� h 7�����5|��[7���>2���u6����1� �*1d���z1 ���7@ ���7�%�~92�%���t6 &�{�4'��vl2'����2d'��RH3�'����5�'�jZ6,(��Ҏ5\(��Gx7�(�>5?3�(� 3�(�� W3 )��d5L*���4P*���4�*� ��4h+�Kӽ5�+�1��4�+�8��4�+�*��4,��Խ5@,�)�c6,�;��4�,����44-�j��7�/���7T1��0R7X1�L��7�1���7�2�i��7<3���7�3���7 4���7p4�ڇx6�4���7�4���t605��d5p6��d5�7�+��7�7�+��7�7�+��7�7���7�7�+��7�7�+��7�7�+��7�7�+��7�7�S��7�7���7�;���74=��d5�=�-{7�>�>a��7�>�>a��7�>�>a��7,?�>a��70?�>a��74?�>a��78?�>a��7<?�>a��7@?�>a��7?�>a��7�?�>a��7�?�>a��7�?�>a��7�?� (�5�?�oX�5�?�a(�5�?�[�t6[email protected]���t6�@�@��7XA�@��7�A�@��7�B�@��7�C�@��7�E�@��7�E�@��7�I�@��7lJ�@��7<K�@��7�K�@��7�K�@��7 L�@��7�M�@��7�N�@��7LO�@��7P�@��7�P�@��7lQ�@��7,R�@��7�R�@��7HS�@��7(T�@��7@V�@��7�V�@��7�V�@��7W�@��7�W�@��7TX�@��7�X�@��7Y�@��7�Y�@��7d[�@��7�[�@��7�[�@��7\�@��74\�@��7�]�@��7 ^�@��7|_�@��7 a�@��7Xa�@��7 b�@��7�b�@��7�b�@��7�c�@��7Pd�@��7�d�@��7le�@��7�e�@��7xf�@��7�f�@��7�g�@��7�h�@��7�i�@��7k�@��7�k�@��7 m�@��7�o�@��7�p�@��7Xq���74s4 0sv��8��0s#g�8ܒ0s���7,�0s ��7D�0s�i�7�0s�i�7Ȕ�0s|��7Ė0sHk�7�0s�i�7�0s�k�7 �0snl�7��0snl�7��0snl�7x�0s�j�7|�0sk�7D�0s�k�7|�0sHk�7�(%/home/was/magma/Kohel/power_series.mg�����(IReturns the echelonized sequence of power series spanning the same space.�0������(�����������X�H�L�����( __sig__0����P���( H�T�����0���(  EchelonSeries�X��(��������P��(;Returns the echelonized basis for the space generated by B.X����(�����������p�H�d�����( __sig__1�� ��p���( H�l�����(��/usr/local/magma-2.5/libsJexamples:galpols:glnzgps:intro:isolgps:matgps:perfgps:pergps:simgps:solgps/tmp     ; �����    J_O @  e̯ gJ ? L� �h��G) 9�RY ����         �5��U�t ! " #  % & ' ( ) * +  , - . ��%�U�t/ ��0 1 234 8˸58˸68˸78˸8 save "valentine.s";600);ile!!)";cter(592,[1,1,6]),2,+1);9%S> r%:� ;f@\�6��H�0p>���8�hp�� ��= D>\ �t*# �4F#L14F0 GQ��u�8'u\�'wQ����������������?�M ����B 0�D" �F8GH �( �� J  �#014FXK 0�4��������8L<stdin>M�4˸N�4˸OtP<Q  phR�lS��(��DT U&V d  WX X������[!����\ ����]"0����| @W� x^*�������������������������������������������� _4˸"�����@W� �a*�������������������������������������������� b4˸�N�defgh��i��j k l m n o p q r s t u v �  IsIrreducible��Classes�xhc�̌h@st1�̖��8(\��X*1h1�C�C�C�C�74T8T<TD �f�f�f<\@\D\H\W�WFF FFlm k�o�p�NyOrderzBase{  FactoredOrder�P{�|  IsVerified}  PowerPrinting~  IsCharacterg DefaultPrecision� SeriesPrintingn�  Precisionting� OutputPrecision�  PrintName� WeightDistribution�� MinimumWeightLowerBound � MatrixPrinting�Maximal� CunninghamStorageLimit�  MinimumWeight� PrimaryInvariantsp�� ��� SecondaryInvariants � SecondarySubmodule�Minimum� MinimumWeightUpperBound� FirstBasicOrbitBound�BSGS�� ˸� ˸�D �|��  HilbertSeriesd  Generators ^  BlocksImage_  BlockSystemb  TensorFactors ! ImprimitiveFlagDHc  TensorBasis�H X �D �< �D �������� ���������<stdin>��<���l�<n�G0u�:h;xG�:8����?�G����P�T' ���������د� �������������<read>�( /home/was/magma/Startup/init.mH/���!/usr/local/magma-2.5/package/spec��>�>����SEA.specP{�� __sig__0eldpackage� <H����Mat�8/usr/local/magma-2.5/package/GrpMat/Smash/extraspecial.m0�00��0�N%Clear.mgm�(� __sig__0malisermK)8��2/usr/local/magma-2.5/package/GrpMat/Smash/larger.m�0�P{��(/usr/local/magma-2.5/package/Code/iaks.m1�1�P1�X���ʸ�2/usr/local/magma-2.5/package/GrpMat/Smash/larger.mP1�P{��8�Compute kernel for homomorphism of semilinear G into cyclic group and write kernel over the larger field; return both representations, the cyclic group C and list of images in C of generators of G�jjj������� __sig__0� ������� WriteOverLargerField�����������8/usr/local/magma-2.5/package/GrpMat/Smash/charpol-test.m�3�P{�58/usr/local/magma-2.5/package/GrpMat/Smash/is-primitive.mP4��4�<True iff G is primitive�4��:��ʸ�8/usr/local/magma-2.5/package/GrpMat/Smash/extraspecial.m�4�P{��MTrue iff G is known to normalise an extrapecial p-group or symplectic 2-group�j������������ �������IsExtraSpecialNormaliser�����������5/usr/local/magma-2.5/package/GrpMat/Smash/functions.m�P{��05/usr/local/magma-2.5/package/GrpMat/Smash/functions.m�P{�� The block system of G�j���������  �� �����YSeedtrix�  BlockSystem���������E���Z��+Z��+Z��+Z��+�J�J�J�J�J�J�J�<G�<G�<G�<G�<G�<G�<G�l �l �l �l �l �l �l ��0˸��0˸� The block system of M�z����������� __sig__1� � ����� __sig__8ters�FReturn prime and exponent of the extraspecial subgroup normalised by M�j��������D4�(� __sig__2� �0����� ExtraSpecialParameters���������,�FReturn prime and exponent of the extraspecial subgroup normalised by M�z��������L�@� __sig__3� �H�����%Return the extraspecial subgroup of M�jj��������d\�P� __sig__4� �X�����%Return the extraspecial subgroup of M�zj��������l�� __sig__5� �h�����The tensor factors of M�z���������|�p� __sig__6� �x�����  TensorFactors���������t�The tensor factors of G�j������������ __sig__7� �������NReturn the change of basis matrix which exhibits the tensor decomposition of M�z ������������� ������� AreProportional�� __sig__1T��  TensorBasis�����������NReturn the change of basis matrix which exhibits the tensor decomposition of G�j ������������ __sig__9� �������  __sig__12sion�&The degree of the field extension of M�z������������� �������  __sig__10� DegreeOfFieldExtension�(D������������&The degree of the field extension of G�j������������  __sig__11� ������� The centralising matrix of M�z ������������ ������ SymmetricTensorFactors� CentralisingMatrix���������� The centralising matrix of Gj �������� �  __sig__13 �����.  __sig__22mutations The Frobenius automorphisms of Mz ��������, �  __sig__14 ����� FrobeniusAutomorphisms ��������  The Frobenius automorphisms of G j ��������4�(   __sig__15  �0����#Return the symmetric tensor factorsz���������D�8  __sig__16 �@���� SymmetricTensorPermutationsLd��������<!Return the symmetric tensor basisz ���������\�P  __sig__17 �X���� SymmetricTensorBasis��������TEReturn the permutations of the factors induced by the generators of Mz���������t�h   __sig__18  �p����R��������8 ��������l #Return the symmetric tensor factors!j�����������"  __sig__19# ������!Return the symmetric tensor basis%j �����������&  __sig__20' ������(EReturn the permutations of the factors induced by the generators of M)j�����������*  __sig__21+ ������, The block permutation group of G-j ������������/ ������0 The block permutation group of M1z �����������2  __sig__233 ������48/usr/local/magma-2.5/package/GrpMat/Smash/is-primitive.m�T�P{�E5/usr/local/magma-2.5/package/GrpMat/Smash/minblocks.m��U�F2/usr/local/magma-2.5/package/GrpMat/Smash/minsub.m.pU��U�U __sig__163Attempt to decompose G wrt normal closure in G of E7j�����������8 __sig__09 ������: SearchForDecomposition;���������=j�����������> __sig__1? ������@True iff M is primitiveAz��������� �B __sig__2C �����D./usr/local/magma-2.5/package/GrpMat/Smash/is.mP{�H3/usr/local/magma-2.5/package/GrpMat/Smash/numbers.mhY��Z�I6/usr/local/magma-2.5/package/GrpMat/Smash/order-test.m�Z�J2/usr/local/magma-2.5/package/GrpMat/Smash/random.mtZ��Z�O __sig__0G0/usr/local/magma-2.5/package/GrpMat/Smash/misc.mm�Z�P{�L6/usr/local/magma-2.5/package/GrpMat/Smash/semilinear.m\�MTrue iff G is semilinear�Nj��������P@04P 0<������ GroupGenerators��qH Q  IsSemiLinearK6/usr/local/magma-2.5/package/GrpMat/Smash/semilinear.mP{��1/usr/local/magma-2.5/package/GrpMat/Smash/smash.m�\�h^��4True iff G is known to be a symmetric tensor product�j�����������H��6� __sig__0� ������� IsSymmetricTensor�� __sig__0STrue iff M is semilinearTz��������X0LV 0T����W6/usr/local/magma-2.5/package/GrpMat/Smash/smash-spec.mP{�X6/usr/local/magma-2.5/package/GrpMat/Smash/smash-spec.mP{�[  BlockNumbers\  BlockSizes]  NmrOfBlocksi DegreeOfFieldExtension!f ExtraSpecialGroupg ExtraSpecialParameters0j  LinearPart } __sig__2()h SemiLinearFlag�a TensorProductFlageExtraSpecialFlag !"#mSymmetricTensorProductFlag�s3 Return G-module generated by sequence of matrices ()oSymmetricTensorPermutationsH�qq �q3d%�  randomSeed./01p SymmetricTensorBasisl FrobeniusAutomorphisms!y __sig__1em*+k CentralisingMatrixn SymmetricTensorFactors!u __sig__0snsorBasiss!� squareDiscriminat !�signum)�sl01�so89� identity8�fieldH�su �sp%�  recognize./�forms7�  generators�q  !"#%&'()*+,-./01tz�����������v �����w. Return G-module generated by set of matrices xz����������z �����{ Return degree of matrix g|k����������~ ����� Return degree of matrices of G �j��������dU�� __sig__3� �����ppMatld�& Return the dimension and field for G �j������( � __sig__4� ����� BasicParameters����������& Return the dimension and field for G � ������80� __sig__5� ,�����3 Return the dimension and field for the G-module M �z������@4� __sig__6� <�����C Return a sequence containing generators of the underlying group G � ��������PD� __sig__7� L�����������������������H�C Return a sequence containing generators of the underlying group G �z��������ph\� __sig__8� d�����8 Return a sequence containing generators of the group G �j��������xl� __sig__9� t�����1/usr/local/magma-2.5/package/GrpMat/Smash/smash.mq�P{��2/usr/local/magma-2.5/package/GrpMat/Smash/tensor.mrhq�Hs��3/usr/local/magma-2.5/package/GrpMat/Tensor/action.m�q�Hs��3/usr/local/magma-2.5/package/GrpMat/Tensor/direct.mr�Hs��;/usr/local/magma-2.5/package/GrpMat/Tensor/factorise-poly.mXr�Hs��7/usr/local/magma-2.5/package/GrpMat/Tensor/find-point.mHs�� ������Mat/�����������;/usr/local/magma-2.5/package/GrpMat/Smash/stabiliser-test.m�s�P{��6/usr/local/magma-2.5/package/GrpMat/Tensor/is-tensor.mv��</usr/local/magma-2.5/package/GrpMat/Tensor/is-projectivity.mt�v��6/usr/local/magma-2.5/package/GrpMat/Tensor/is-tensor.mv�� True iff G is a tensor product�j����������� __sig__0�� ��������������t�k T� IsTensor�7/usr/local/magma-2.5/package/GrpMat/Tensor/find-point.m�4�Decide whether or not the collection of matrices X is composed of k x k blocks which differ only by scalars; if so, return true and the decomposition of each matrix, else false������������� SemiLinearGroup��dmallerFiel�����������,�Decide whether or not the matrix X is composed of k x k blocks which differ only by scalars; if so, return true and the decomposition, else false������������� ������� IsProportional�����������5/usr/local/magma-2.5/package/GrpMat/Tensor/groebner.m�P{��2/usr/local/magma-2.5/package/GrpMat/Tensor/local.m�z�h}��2/usr/local/magma-2.5/package/GrpMat/Tensor/order.m5�z�h}�� ������ __sig__0� lx�����2/usr/local/magma-2.5/package/GrpMat/Tensor/pdash.m8{�h}��3/usr/local/magma-2.5/package/GrpMat/Tensor/plocal.m�{�h}��7/usr/local/magma-2.5/package/GrpMat/Tensor/stabiliser.mh}��(/usr/local/magma-2.5/package/GrpAb/hom.mppd(|�x|�����ʸ�?/usr/local/magma-2.5/package/GrpMat/SmallerField/SmallerField.mh}��?/usr/local/magma-2.5/package/GrpMat/SmallerField/SmallerField.mh}�� True iff M is a tensor product�z�������� �� __sig__1�3/usr/local/magma-2.5/package/GrpMat/Tensor/jordan.m~�P��2/usr/local/magma-2.5/package/GrpMat/Misc/semilin.m�~�H���1The semilinear extension of G over the subfield Sd5�qjj��������|lpH��6� __sig__0/  IsLinearGroup�0���������# RecognizeClassicalH' __sig__1P���2/usr/local/magma-2.5/package/GrpMat/Misc/rewrite.m��H���2/usr/local/magma-2.5/package/GrpMat/Misc/rewrite.m50��H���qjj���������d5� __sig__08��������"kDoes G have an equivalent representation over a subfield? If so, return true and equivalent representation.�jj������\L<@� IsOverSmallerField���������D�&wDoes G have an equivalent representation over a subfield of degree d? If so, return true and equivalent representation.�jj������d<X� __sig__1� <�����2/usr/local/magma-2.5/package/GrpMat/Misc/semilin.m��P��Entropyund "jUse the Niemeyer-Praeger algorithm to test whether <grp> contains the corresponding classical group Omega. j���������x|! __sig__0" x�����,1 LReturn true if grp is known to be contained in GSp(d,q) and contains Sp(d,q)������������%rReturn the type if <grp> is known to be a classical matrix group in its natural representation and false otherwise�&j���������x�( x�����)  ClassicalType*���������- __sig__2�<�Given a group G of d by d matrices over a finite field E having degree e and a subfield F of E having degree f, write the matrices of G as d*e/f by d*e/f matrices over F and return the group and the isomorphism� WriteOverSmallerField�����������=/usr/local/magma-2.5/package/GrpMat/ClassicalRec/attributes.m�P��</usr/local/magma-2.5/package/GrpMat/ClassicalRec/classical.m�P��  containsSL isSL'isGL/  primeField89  statisticBC�  randomProcessL isCRecSL   smallRandom()ranges1 dualGenerators; abstractIdentityFGHI  printLevel� matmap  orderSL%&'  transvection1 transvectionBasis<= useTransvectionGHIextPQ  moreAbstractGens�   abstractGroupspanningConjugatingElements"# spanningBasisInverse/  spanningBasis9 rewritingRulesC  detOrdersLMorderU  abstractSeed�����  almostElement�root�  rootDeterminant�� images�  </usr/local/magma-2.5/package/GrpMat/ClassicalRec/classical.m�����  abstractGenerators�j��������� �/� � ����;� NumberOfInvariantFormsG���������� __sig__1_+KReturn true if grp is known to be contained in GL(d,q) and contains SL(d,q)�,j���������x��. x�����I ClassicalForms�J�������� ; IsOrthogonalGroupH<���������2j���������x�C3 __sig__3LM4x�����XYZ�j���������P�D�� __sig__2 X p � �L����� �NumberOfAntisymmetricForms\ t ���������H� eThe symmetric and anti-symmetric invariant forms of G (the first symmetric form is positive definite)(�(/usr/local/magma-2.5/package/Lat/genus.m�������2The genus of L (the non-isometric neighbours of L)�����������Y���<� __sig__0P� ������h�SformsD8/usr/local/magma-2.5/package/GrpMat/ClassicalRec/forms.m�����EDreturns record storing the classical forms of the matrix group <grp>d5Fj����������   �G __sig__0d5H  �����d5a ScalarsSymmetricBilinearFormb��������x U ScalarsSymplecticFormV��������H Lj ��������8  , (N  4 ����O SymplecticFormP��������0 _ __sig__4Form�(M148alentin x5 IsSymplecticGroup9 __sig__4�( epsDeterminant;;oQ8If <grp> preserves a symplectic form, return the scalars6���������7tReturns true if grp is known to be contained in GO^e(d,q) and contains the corresponding subgroup Omega of SO^e(d,q)8j���������x�: x�����K5If <grp> preserves a symplectic form, return the form�M __sig__1��� =HReturn true if grp is known to be contained GU(d,q) and contains SU(d,q)>j�������� x�? __sig__5@ x�����A IsUnitaryGroupB���������C8/usr/local/magma-2.5/package/GrpMat/ClassicalRec/forms.m؟�P��:�An abelian group A isomorphic to Hom(G, H) (where G and H are finite abelian groups), together with the transfer map t which, given an element of A, returns the corresponding group homomorphism from G to H�����������    � __sig__0��d5�   �����<�An abelian group A isomorphic to Hom(G, H) (where G and H are finite abelian PC groups), together with the transfer map t which, given an element of A, returns the corresponding group homomorphism from G to Hd5����������   d5� __sig__10�  �����(�A sequence of (Z-module) generators of the set of all homomorphims from the finite abelian group G to the finite abelian group H�� __sig__2Rj ��������P  D S __sig__2T  L ����c4If <grp> preserves a quadratic form, return the form�f  � ���� W=If <grp> preserves a symmetric bilinear form, return the formXj ��������h  \ Y __sig__3Z  d ����[ SymmetricBilinearForm\�������� e __sig__5ilinearForm]?If <grp> preserves a symmetric biliner form, return the scalars^j ���������  t   | ����} FormType��~��������� �������e/bkab.mmq ScalarsUnitaryFormHr��������� �Modulesrsdj ���������  � o __sig__7ormg7If <grp> preserves a quadratic form, return the scalarshj ���������  � i __sig__6j  � ����k ScalarsQuadraticForml��������� m5If <grp> preserves a unitary form, return the scalarsnj ���������  � p  � ����6/usr/local/magma-2.5/package/GrpMat/ClassicalRec/ppd.m8��� (/usr/local/magma-2.5/package/GrpAb/hom.mica��8������ʸs2If <grp> preserves a unitary form, return the formtj ���������  � u __sig__8v  � ����w  UnitaryFormx��������� y4If <grp> preserves a classical form, return its typezj���������  � { __sig__9|  � ����� SrivastavaCode����������� � | � ����� � AllHomomorphismsH���������T � __sig__5 � � � 8 �(/usr/local/magma-2.5/package/Code/cons.m��p��� __sig__1tavaCode�bhGiven sequences alpha = [a_1, ..., a_n], W = [w_1, ..., w_s], and Z = [z_1, ... z_k] of elements from the extension field K of the finite field S, such that the elements of alpha and Z are non-zero and the n + s elements of alpha and W are distinct, together with an integer t > 0, construct the generalized Srivastava code of parameters alpha, W, Z, t, over S��q�r�r�rH��������� | � � __sig__0� | � ���� �GeneralizedSrivastavaCode(���������� �q�r�rH��������� | � �� | � �����  GabidulinCode���������� � __sig__3�����������D 4  ( �  0 �����  HomGenerators���������, �(�A sequence of (Z-module) generators of the set of all homomorphims from the finite abelian PC group G to the finite abelian PC group H���������L  @ � __sig__3�  H ����� \A sequence of all homomorphims from the finite abelian group G to the finite abelian group H�����������l \  P � __sig__4�  X ������r�r�rH��������� | � �� __sig__2 X p 6Primes� � bA sequence of all homomorphims from the finite abelian PC group G to the finite abelian PC group H���������t  h �  p �����(/usr/local/magma-2.5/package/Code/cons.mк�P��(/usr/local/magma-2.5/package/Code/iaks.m������ __sig__1ation�&xGenerate the homomorphic image of Code in a SubField, i.e., each symbol in Code is represented as sequence over SubField�H��6�qHH��������� � � � __sig__0�d5� � � ������ SubfieldRepresentation���������� �<Construct the concatenated code from OuterCode and InnerCoded5�HHH�������� � � �  ����0�qH��������\ 4 P �� __sig__1 X p �4 X �����  , �rHH�������� �  � __sig__2�V:Given sequences alpha = [a_1, ..., a_n] and W = [w_1, ..., w_s] of elements from the extension field K of the finite field S, such that the elements of alpha are non-zero and the n + s elements of alpha and W are distinct, together with an integer mu, construct a Srivastava code of parameters alpha, W, mu, over S� ConcatenatedCode� ��������� � � ����� Relat� �  ChienChoyCode ���������� �  MaxOrthPCheckackage�Aforms�ʸ�@�Given alpha = [a_1,...a_n], and w = [w_1,...w_s] a sequence of n+s distincts elements of GF(q), z, a sequence of n nonzero elements of GF(q), and t > 0, construct the Gabidulin MDS code of parameters alpha, w, z, t over GF(q)�ZMGiven polynomials P and G over a finite field F, an integer n > 1, together with a subfield S of F, such that n is coprime to the cardinality of S, F is the splitting field of x^n - 1 over S, P and G are coprime to x^n - 1 and both have degree less than n, construct the Chien-Choy generalised BCH code with parameters P, G, n over S�qH��������� | � � | � �����*�Construct the Justensen code from RSCode and a primitive element alpha of the alphabet of RSCode the powers of the innercode are shifted by k< �H��������� � � ��  SimplexCode����������� �(/usr/local/magma-2.5/package/Code/fire.m�������(/usr/local/magma-2.5/package/Code/fire.md5(������*�Given a polynomial h in GF(q)[X], a nonnegative integer s, constructs a Fire code of length n of generator polynomial h*(X^s - 1) over GF(q)�H��������� � � d5� __sig__0�� � � ���� LReturn the Elias asymptotic upper bound for delta in [0, 1] over the field K� VqV��������<  0 d �*/usr/local/magma-2.5/package/Code/random.m�������HReturn a random linear code of length n and dimension k over the field K �qH�������� � � � ZeroCode�����������< �'Return the [n,n,1] universe code over F8  � �  JustensenCode��������� �*/usr/local/magma-2.5/package/Code/simple.m��P�� */usr/local/magma-2.5/package/Code/simple.mP��P��#Return the [n,0,n] zero code over F�qH��������D 4 8 � __sig__0� 4 @ ����� EvenWeightCode����������� �+/usr/local/magma-2.5/package/Code/simplex.m��p��� __sig__0 x � � RepetitionCode ���������l �)Return the [n,n-1,2] zero-sum code over F� � �  UniverseCode���������T �#Return the [n,1,n] zero code over F�qH��������t 4 h � __sig__2� 4 p �����+/usr/local/magma-2.5/package/Code/simplex.m���x���0Return the [2^r-1,r,2^(r-1)] simplex binary code � � � � � ����, �qH��������� 4 � � __sig__3� 4 � �����  ZeroSumCode���������� �0Return the [n,n-1,2] even-weight code over GF(2)�H��������� 4 � � __sig__4� 4 � ����   EliasBound�� ��������  __sig__4inearBound __sig__1 � �   8 ���� � FireCode� ���������� �*/usr/local/magma-2.5/package/Code/random.m��p�� __sig__0 � � .)/usr/local/magma-2.5/package/Code/bound.mp��p��vReturn the upper Elias bound for the cardinality of a largest code of length n and minimum distance d over the field Kq��������   �d5 __sig__0,   ����" GilbertVarshamovAsymptoticBound#��������| '  � ����8 ( GriesmerLengthBoundx � Modi p  GilbertVarshamovBound��������L q��������l   �  EliasAsymptoticBound���5��������4 (�Return the Gilbert-Varshamov lower bound for the cardinality of a largest code of length n and minimum distance d over the field K�~ z2q��������T  H  � ���� RandomLinearCode��������� )/usr/local/magma-2.5/package/Code/bound.m��P��+/usr/local/magma-2.5/package/Code/entropy.mX������ UComputes the entropy function for a real number in the interval [0,1-1/q], where q=#Sd5�qVV��������L<@�� __sig__0d5� <H�����d5�*�Return the best known lower bound and upper bound for the cardinality of a largest code of length n and minimum distance d over the field K� < �q��������� � __sig__0 � � � <����(� InformationRate����������\�)/usr/local/magma-2.5/package/Code/ispow.m�������)/usr/local/magma-2.5/package/Code/ispow.mo6�������MTest if Q is a power of q>0, then return true and the exponent or false and 0S�Q7��������tx� __sig__0y z2� t����� __sig__2  P ����"kReturn the Griesmer lower bound of the length of a linear code of dimension k and minimum distance d over K( @ % q���������  � D \ t (�Return the Gilbert-Varshamov lower bound of the size of a largest linear code of length n and minimum distance d over the field K __sig__3  h ���� GilbertVarshamovLinearBound ��������d & __sig__5symptoticBound pReturn the Gilbert-Varshamov asymptotic lower bound of the information rate for delta in [0, 1] over the field K VqV���������  x !  � ����:  GriesmerBound�;��������� <"nReturn the Johnson upper bound for the cardinality of a largest binary code of length n and minimum distance d@ .  IsOptimal� /��������� 8 __sig__8ightBound1q���������  � � )��������� * SReturn true if the linear code is optimal for the Griesmer bound, else return false+H���������  � , __sig__6-  � ����=���������  � �> __sig__9 X p ?  � ����� @  JohnsonBound SQroctt0tCompute the Griesmer upper bound of the minimum weight of a linear code of length n and dimension k over the field K2 __sig__73  � ����4GriesmerMinimumWeightBound5��������� 6"mReturn the Griesmer upper bound of the cardinality of a linear code of length n and minimum distance d over K7q���������  � 9  � ����RMcElieceEtAlAsymptoticBoundHS��������< TpReturn the Plotkin upper bound of the cardinality of a largest code of length n and dimension d over the field K � � F IsNearlyPerfect G�������� H(wReturn the Levenstein upper bound of the cardinality of a largest code of length n, minimum distance d over the field K� � � � A��������� BCReturn true if the binary code is nearly perfect, else return falseCH��������   D  __sig__10E   ����Uq��������\  P �W  X ����� X  PlotkinBound � Y��������T Iq��������,  J  __sig__11K  ( ����L LevenshteinBoundM�������� V  __sig__13oticBoundNtReturn the McEliece-Rodemich-Rumsey-Welch asymptotic upper bound of the binary information rate for delta in [0, 1].OVV��������D  8 P  __sig__12Q  @ ����j SingletonBound�k��������� t  __sig__18cBound l"mReturn the Singleton asymptotic upper bound of the information rate for delta in [0, 1] over any finite field0 ^  IsEquidistant _��������l h  __sig__16ound� "fReturn the Plotkin asymptotic upper bound of the information rate for delta in [0, 1] over the field K0 H  Z9Return true if the code is equidistant, else return false[H��������t  h \  __sig__14]  p ����mVV���������  � �n  __sig__17X p o  � ����� pSingletonAsymptoticBound \ t u  � ����� aVqV���������  � b  __sig__15c  � ����d PlotkinAsymptoticBounde��������� fvReturn the Singleton upper bound of the cardinality of a largest code of length n, minimum distance d over the field Kgq���������  � i  � ����� HammingAsymptoticBound ��������� �&zReturn the "Van Lint" lower bound to the size of a largest linear code of length n and minimum distance d over the field K� � �  __sig__22� � � � v  SphereVolume w��������� x(�Return the Hamming spheres packing upper bound of the cardinality of a largest codes of length n and dimension d over the field K� �Nendskmq��������� rJComputes the volume of the closed ball of radius r in the vector space K^nsq���������  � �q��������, ��  (����� ����������+/usr/local/magma-2.5/package/Code/entropy.m���P��yq���������  � z  __sig__19{  � ����| SpheresPackingBound}��������� ~5Return true if C is a perfect code, else return falseH�������� � �  __sig__20�  �����  VanLintBoundd� fReturn the Hamming asymptotic upper bound of the information rate for delta in [0, 1] over the field K�VqV�������� �  __sig__21�  �����  IsPowerOf�����������|�'/usr/local/magma-2.5/package/Code/bkb.m@��'/usr/local/magma-2.5/package/Code/bkb.m@�� __sig__0ightCode�������0Ph���������D�-Computes the information rate of a given codel �HV��������d<Xp � __sig__1 H  �(/usr/local/magma-2.5/package/Code/kraw.mX�� �� RGiven a polynomial P over the Rational Field, return the polynomial Binomial(P, l)��d5������������� �������  LeeWeight�����������T�+/usr/local/magma-2.5/package/Code/mattsol.m���+/usr/local/magma-2.5/package/Code/mattsol.mX��� __sig__1nsform � __sig__0Ph�de�&wReturn the inverse Krawchouk transform of the polynomial A over the rational field with respect to the vector space K^n� � � �q����������<�0� � __sig__3 � � � �8����� __sig__1� �o6� ������ KrawchoukPolynomial0 ����������oReturn the Krawchouk transform of the polynomial a over the rational field with respect to the vector space K^n��q���������� __sig__2P� � ����l � KrawchoukTransform� � ��������� � InverseKrawchouk � � � __sig__0���BKBdsCardLargestLinearCodex������������&xReturn the best known bounds to the maximal minimum weight of a linear code of length n and dimension k over the field K  0 � __sig__2 0� BKBdsCardLargestCode�����������*�Compute the best known lowerand upper bounds to the cardinality of a largest linear code of length n and minimum weight d over the field K�q���������� __sig__1� ������� BinomialPolynomial������������ QReturn the Krawchouk polynomial of parameter k and n in K over the Rational Field� � �q�������� �l�q���������� ������� BKBdsMaxMinimumWeight�����������(/usr/local/magma-2.5/package/Code/kraw.m��P��'/usr/local/magma-2.5/package/Code/lee.m���8Return the Lee weight of the vector v over a prime field��d5�u��������\LP� LX�����,/usr/local/magma-2.5/package/Code/decodeml.m����>Return a maximal set of equations orthogonal for each position� �H������������ ���������� __sig__1���� InverseMattsonSolomonTransform������������*/usr/local/magma-2.5/package/Code/isproj.m0� ��*/usr/local/magma-2.5/package/Code/isproj.mx� ��(Return true iff the code C is projective���*�Given a, a polynomial over a finite field containing a primitive n-th root of unity, return the Mattson-Solomon transform of parameter n ���������|lpp� lx����� MattsonSolomonTransform���������t� __sig__0omonTransform��,�Given A, a polynomial over a finite field containing a primitive n-th root of unity, return the inverse Mattson-Solomon transform of parameter n � � ����������l�D���������4�'/usr/local/magma-2.5/package/Code/lee.mP��H��������@0���� �������d5�  IsProjective�����������(/usr/local/magma-2.5/package/Code/bkab.md5�� ��(/usr/local/magma-2.5/package/Code/bkab.m�� ��sReturn the best known asymptotic lower and upper bound of the information rate for delta in [0, 1] over the field K�d5�VqVV���������� __sig__0d5� ������� ���������<,  �   (����� '/usr/local/magma-2.5/package/Lat/code.m� � '/usr/local/magma-2.5/package/Lat/code.m� �9The lattice constructed from code C according to method Sd�� BKABdsInfoRate~ z2����������� __sig__02.5/package�(/usr/local/magma-2.5/package/Lat/genus.m �P9�\1��,/usr/local/magma-2.5/package/Code/decodeml.m � �� __sig__0� l�����<�Decoder for binary codes. Given a list of received vectors, return a two sequences. The second contains the corrected vectors, the first, boolean values saying if the decoder could retrieve the original codeword, D �uH������ ��H���������D48 __sig__0� 4@����'/usr/local/magma-2.5/package/Lat/cons.m &�2'/usr/local/magma-2.5/package/Lat/cons.m &� The sum of lattices L1 and L2�������������\LP __sig__0��d5 LX���� PThe direct sum of lattices L1 and L2 (with orthogonal sum of the inner products)������������lL( __sig__1� Lh����  PThe direct sum of lattices L1 and L2 (with orthogonal sum of the inner products)� ������������|Lp  __sig__2   Lx����0   OrthogonalSum�!��������t" OThe direct sum of the lattices in Q (with orthogonal sum of the inner products)�B z2#������������-�L�� __sig__3 z2%L����� DecodeML����������+/usr/local/magma-2.5/package/Lat/standard.m�&��'�+/usr/local/magma-2.5/package/Lat/standard.m�&��'�%The standard lattice given by S and n�  __sig__0�JDensity� �����K�������� LThe centre density of L���M�����������<L0� N  __sig__12� � O L8����� P  Return s * LR  __sig__13��>  DualQuotient ?���������@&The p-neighbour of L with respect to v� A����������� L0 B  __sig__100C L������)/usr/local/magma-2.5/package/Lat/lambda.md5�*�p1�� The laminated lattice of rank n�����������D48� __sig__0� 4@�����  DeepHoles������������� The covering radius of lattice L � 8 �����������T� � __sig__4 � � � T�����@�%MySorte� __sig__2� � T�����P�����������The deep holes of lattice Lx � �����������T�0� __sig__3(@� T�������(�The Voronoi cell of lattice L, as the vertices (a sequence of vectors), the set of edges, and the planes (a sequence of vectors)p������pTd� Tl������  VoronoiCell���������h� The Voronoi graph of lattice L� __sig__1& OThe direct sum of the lattices in Q (with orthogonal sum of the inner products)'������������L�( __sig__4) L�����*(The tensor product of lattices L1 and L2+�����������3�L�, __sig__5- L�����. The exterior square of lattice L/�����������L�0 __sig__61 L�����2!The symmetric square of lattice L3�����������L�4 __sig__75 L�����6 The dual Lattice of L7�����������.�L�8 __sig__89 L�����: ^The dual quotient L#/L of L, together with the dual L# and the natural epimorphism f:L -> L#/L;��������L�< __sig__9= L�����Q����������TLL@�S LH����� T  Return L * s � U�����������\LP� V  __sig__14��W LX�����D  NeighbourE��������F(The centre density of L, in real field KG�������������4LH  __sig__11I L ����� LaminatedLattice� ���������<�(/usr/local/magma-2.5/package/Lat/holes.m @;�p<�� (/usr/local/magma-2.5/package/Lat/holes.m �;�p<��  DirichletVert� � DirichletPlanes���� __sig__0���Holes(~  MinkowskiMap ����������4Apply the Minkowski map to the element a of an order � ���������� L� �  __sig__26��� L�����X  Return L / sY����������tlLZ  __sig__15[ Lh����\  Return L / s]����������|Lp^  __sig__16_ Lx����)The ring of coefficients of the lattice La������������^�L�b  __sig__17c L�����d CoordinateRinge���������f TApply Seysen reduction to L, returning a new lattice and the transformation matrix Tg�� �������L�h  __sig__18i L�����j RApply pair-reduction to L, returning a new lattice and the transformation matrix Tk�� �������L�l  __sig__19m L�����n&xA lattice with Gram matrix equal to L but with a diagonal inner product matrix,together with the transformation matrix To�� �������L�p  __sig__20q L�����rFApply the Minkowski map to the element a of the algebraic number fieldith Es����������L����u L�����dG�|  __sig__24v XA lattice with Gram matrix equal to L over real field K but with Euclidean inner productw��������������L�x  __sig__22y L�����t  __sig__25��z bA lattice with Gram matrix equal to L over the default real field but with Euclidean inner product{�����������L�} L������������������T|�� T������ �  VoronoiGraph � ����������� The holes of lattice L� � ����������T���,�(/usr/local/magma-2.5/package/Lat/kappa.mPJ�P��(/usr/local/magma-2.5/package/Lat/kappa.m�J�P�� The Kappa lattice K_n����������� __sig__0�  �����  KappaLattice��������� �)/usr/local/magma-2.5/package/Lat/lambda.m8L�P��B*/usr/local/magma-2.5/package/Lat/iterate.m�L�S�� Nsforms�� Naforms��d5� Ncenends � EndomorphismRing*+�CentreOfEndomorphismRing89� __sig__3tricForms�FThe number of invariant (both symmetric and anti-symmetric) forms of GZ� __sig__0cfgh# < � �����D __sig__7��E <  �����F  ReplacePrimes G�������� H.Check the correctness of the soluble quotient.� IO��������, < �J __sig__8,�K < ( �����L SQ_check��M�������� P __sig__9 � �   Initialize8%��������� & fInitialize a SQProc for a given quotient epi:F -> G without any information about the relevant primes. ZXR !"#%&'()*+,-./0123456789:;<=>?@ABCDEFGHIJKLMNOPQRSTUVW8XYZ[\]^_abcfghijklm�,The number of symmetric invariant forms of G��j���������8�,� �4����<� AntisymmetricForms� �����������9n invariant forms of G (either symmetric or antisymmetric ( �j������������ � __sig__6��� �������� __sig__7��� InvariantForms ���������� RThe symmetric invariant forms of G (the first symmetric form is positive definite)� � �j�����������th� __sig__4�� �|����(DataT� NumberOfSymmetricForms0 ���������0�1The number of anti-symmetric invariant forms of GHt�*/usr/local/magma-2.5/package/Lat/iterate.mpX�P�h��j������������j �������d5k  Sublatticesl���������m&}The G-invariant sublattices of the natural lattice L of G which are not contained in p.L and whose index in L is a power of pd5nj�������������o __sig__1d5p ������ d5qMRepresentatives for the G-invariant sublattices of the natural lattice L of Grj������������0s __sig__2�t ������u,�The G-invariant sublattices of the G-lattice L which are not contained in p.L for any p in Q and whose index in L is a product of elements of Q�v���������������w __sig__3�Z z2x ������yrThe G-invariant sublattices of the G-lattice L which are not contained in p.L and whose index in L is a power of p�z�������������{ __sig__4�If�3| ������}BRepresentatives for the G-invariant sublattices of the G-lattice L��'7~������������j���������h�\� �d����� an symmetric invariant forms of G (the first symmetric form is positive definite if n is non-zero)� �j����������� � ������4 �%n anti-symmetric invariant forms of G�� j��������(���Xp�� SymmetricForms���������x�'The anti-symmetric invariant forms of G�j������������� __sig__5� ������? CentralEndomorphisms @���������AKThe centre of the endomorphism ring of the integral/rational matrix group Gx � Bj ��������H�0C  __sig__26(@D � �����ECentreOfEndomorphismRing�t��#  Endomorphisms ���������%=The endomorphism ring of the integral/rational matrix group G( &j ������������� '  __sig__20��( �������1  __sig__22phismRingLimitsp� __sig__8� ������  __sig__12ricForms�FThe number of invariant (both symmetric and anti-symmetric) forms of G�������������� __sig__9� �������,The number of symmetric invariant forms of G��������������  __sig__10� �������1The number of anti-symmetric invariant forms of G�������������  __sig__11 ������ eThe symmetric and anti-symmetric invariant forms of G (the first symmetric form is positive definite)���������8� � ���� RThe symmetric invariant forms of G (the first symmetric form is positive definite)���������H �  __sig__13 � ���� 'The anti-symmetric invariant forms of G ���������X0�   __sig__14  �,���� 9n invariant forms of G (either symmetric or antisymmetric���������@�4  __sig__15 �<���� an symmetric invariant forms of G (the first symmetric form is positive definite if n is non-zero)���������P�D  __sig__16 �L����%n anti-symmetric invariant forms of G����������T  __sig__17 �\���� The Bravais group of Gjj��������p�d  __sig__18  �l����   BravaisGroup ��������h  ^N independent endomorphisms from the endomorphism ring of the integral/rational matrix group G j�����������|!  __sig__19" ������F��������[  __sig__31fEndomorphismRingG \The dimension of the centre of the endomorphism ring of the integral/rational matrix group G Hj ��������X(� �J ������K#DimensionOfCentreOfEndomorphismRing@X)NThe dimension of the endomorphism ring of the integral/rational matrix group G*j ������������+  __sig__21, ������-DimensionOfEndomorphismRing.���������/IN independent endomorphisms from the endomorphism ring of the G-lattice L0������������2 ������3(The endomorphism ring of the G-lattice L4� �����������5  __sig__236 ������I  __sig__27smRingng79The dimension of the endomorphism ring of the G-lattice L8� �����������9  __sig__24: ������; fN independent central endomorphisms from the endomorphism ring of the integral/rational matrix group G<j��������8���=  __sig__25> ������c DimensionOfHom�d���������e)/usr/local/magma-2.5/package/Lat/sublat.m�|�~�f )/usr/local/magma-2.5/package/Lat/sublat.m�|�~�g.�The G-invariant sublattices of the natural lattice L of G which are not contained in p.L for any p in Q and whose index in L is a product of elements of Q��i __sig__0��L�������� M QN independent central endomorphisms from the endomorphism ring of the G-lattice L� � N���������@�4 O  __sig__28PhP �<�����Q6The centre of the endomorphism ring of the G-lattice L��1 Returns True if the ring map phi is surjective. X�����������.�H�� � __sig__3 � � � H����� �����������H��� __sig__4�,� H�����x� Implicitization@R� ��������P�DS  __sig__29T �L����UGThe dimension of the centre of the endomorphism ring of the G-lattice LV� ���������TW  __sig__30X �\����YGA non-trivial homomorphism from G1 to G2, or zero if one does not existZD��������p�d\ �l����]  Homomorphism^��������h_4The dimension of the homomorphism module Hom(G1, G2)jj����������|a  __sig__32b ������� JacobianSequence� ����������� OReturns the ideal generated by all first partial derivatives of the polynomial.� � � ����������� �h� __sig__1�� � ����(�  JacobianIdeal�� __sig__2���&/usr/local/magma-2.5/package/Mod/aut.mH���The automorphism group of M��zj���������70 � __sig__0�H��6�  ,�����+/usr/local/magma-2.5/package/SeqEnum/sort.mP��H���+/usr/local/magma-2.5/package/SeqEnum/sort.m���H���@Sort the integer sequence Q using the diminishing increment sortd5�X������PH8<d5� __sig__0 � 8D�����@Sort the integer sequence Q using the diminishing increment sort4�X����������X8L� __sig__1� 8T����� �@Sort the integer sequence Q using the diminishing increment sort����X����������h8\� __sig__2���63� 8d�����Return true iff I equals Jl/RingMaps.m z2�0/usr/local/magma-2.5/package/SeqEnum/eseq_misc.m���H���0/usr/local/magma-2.5/package/SeqEnum/eseq_misc.mЍ�H��� The join of all elements of S ��-Z6��00����������pt� p|������ The join of all elements of S  __sig__5� �������*/usr/local/magma-2.5/package/Mod/irrmods.m���P��*/usr/local/magma-2.5/package/Mod/irrmods.m��P��@�Given a G-module M for a group G over a finite field, construct NumIrrs irreducible modules for G by forming tensor products of M and consequent irreducibles provided that the degree of a tensor product does not exceed DimLim�>�������� � __sig__0� �����&/usr/local/magma-2.5/package/Mod/aut.mP��./usr/local/magma-2.5/package/SetMulti/minmax.mؙ�� The minimum of the elements of S����������������� __sig__0 � ������� The maximum of the elements of S\���������������� __sig__1� ��������//usr/local/magma-2.5/package/RngMPol/IdealOps.mؙ��//usr/local/magma-2.5/package/RngMPol/IdealOps.mؙ��HReturns the sequence of all first partial derivatives of the polynomial.d5�����������������5� __sig__0�� ������� JacobianMatrix����������0�//usr/local/magma-2.5/package/RngMPol/RingMaps.m(��� //usr/local/magma-2.5/package/RngMPol/RingMaps.m(���'Display a map between polynomial rings.� 0�����������XHLt� __sig__0�t� DisplayPolyMap����������(�Returns the ideal generated by the partial derivatives of each polynomial in the sequence, by each indeterminate in its parent ring.��4�������������^(� �� ������ __sig__3��00���������p�� __sig__1� p������./usr/local/magma-2.5/package/SetMulti/reduct.mP��./usr/local/magma-2.5/package/SetMulti/reduct.mP��6The sum (counting multiplicities) of all elements of S���������������� __sig__0� �������:The product (counting multiplicities) of all elements of S���������������� __sig__1� �������./usr/local/magma-2.5/package/SetMulti/minmax.mP��"Return true iff I is a subset of J��SS���������S���� __sig__1��d5� ������� Return the ideal I + J�SSS��������X.���� __sig__2�� ������� Return the ideal I * Jd5�SSS��������-��� __sig__3d5 ������� Return the ideal I ^ nSS���������,� __sig__4 � �����(�Return the colon ideal I:J (or ideal quotient of I by J), consisting of the polynomials f of P such that f * g is in I for all g in J 3SSS�������� �� __sig__5�63 � ����b z2 -Return the intersection of the ideals I and J SSS���������S0�  __sig__6  �,����H 9Return true iff I is not equal to the whole quotient ringPS��������@�4 __sig__7 �<����lReturn true iff I is a maximal idealS��������P�D�,�Returns the matrix with (i,j)'th entry the partial derivative of the i'th polynomial in the list with the j'th indeterminate of its parent ring.���j���������^8�,� �4�����  PolyMapKernel����������h�M Returns True if the polynomial the_poly is in the image of the ring map phi. �����������H|� � __sig__2�� H������  IsInImage������������ HT�������������P�E Creates the kernel of a homomorphism "phi" between polynomial rings.����������pHd� __sig__1� Hl����[ GaussianIntegerRing� \��������]MThe ring of Gaussian integers (the maximal order of the quadratic field Q(i))� ^����������Xt0_ __sig__1(@ X|�����a GaussianIntegersl�b��������x�����������(/usr/local/magma-2.5/package/HC/jac_hc.m���-��ʸ�:/usr/local/magma-2.5/package/RngMPolRes/AffineAlgebraOps.m��x��� :/usr/local/magma-2.5/package/RngMPolRes/AffineAlgebraOps.mX��x���SS��������<A���t� __sig__0�t�������\t��D� Assumes that phi is a parametrization of a variety defined by equations y_i = f_i(x_i) where the x_i are in the CODOMAIN of phi and the y_i are in the DOMAIN of phi. Returns the smallest variety containing the points defined by these equations.k __sig__3Ring�c VThe ring of Eisenstein integers (the maximal order of the quadratic field Q(Sqrt(-3)))� d����������X�� e __sig__2 | � f X������i VThe ring of Eisenstein integers (the maximal order of the quadratic field Q(Sqrt(-3)))�l X�����2'/usr/local/magma-2.5/package/Rng/comp.m���3*The completion of field K at prime ideal P�4 ������������6 �������7The completion of R at prime ideal Pd58P������������9 __sig__1d5: ������d5;'/usr/local/magma-2.5/package/Rng/frac.m���<'/usr/local/magma-2.5/package/Rng/frac.m���=6The fraction field R/I where I is a maximal ideal of R(>����������? __sig__0h@ ������A6The fraction field R/I where I is a maximal ideal of R6B����������C __sig__1 z2D � ����Z z2E6The fraction field R/I where I is a maximal ideal of R�F���������� �G __sig__2�u z2H � ����I&/usr/local/magma-2.5/package/Rng/loc.m���J&/usr/local/magma-2.5/package/Rng/loc.m���K&The localization of R at prime ideal PL��������H8(,M __sig__0N (4����lO  Localization�P��������0Q&The localization of R at prime ideal PS __sig__1 __sig__8 �L����Return true iff I is a primary idealS���������T __sig__9 �\���� True iff the ideal I is radicalS��������p�d  __sig__10  �l���� "Return whether I is the zero ideal S���������.��t   __sig__11  �|����!p�Given an ideal I of a quotient ring P=R[x_1, ..., x_n] / J, where J is an ideal of R[x_1, ..., x_n], with coefficient ring R, together with a ring S, construct the ideal J of the polynomial ring Q=S[x_1, ..., x_n]/J obtained by coercing the coefficients of the elements of the basis of I into S. It is necessary that all elements of the old coefficient ring R can be automatically coerced into the new coefficient ring S"��SS�����������#  __sig__12 ������% The radical of I&SS�����������'  __sig__13( ������5 __sig__0ion)The primary decomposition of ideal I*S���������+  __sig__14, ������-"lThe (prime) decomposition of the radical of I; note that this may be different to the associated primes of I.S���������/  __sig__150 ������1'/usr/local/magma-2.5/package/Rng/comp.mP�V*/usr/local/magma-2.5/package/RngInt/misc.m0�����WMThe ring of Gaussian integers (the maximal order of the quadratic field Q(i))�X���������hX\Y __sig__0��d5Z Xd����� ThueSolveInexact� �������������������H@�4� � __sig__2 � � � �<����� �"gTrue and an order whose defining polynomial is better, or false if such a polynomial could not be found���������XP�D�j����������X��n���������o"jThe integer nearest to x (rounded towards infinity) if x real; otherwise the nearest Gaussian integer to x� � pV�����������X�(q __sig__4l�r X������"mTrue and a number field whose defining polynomial is better, or false if such a polynomial could not be found����������8(� d � __sig__1 \ t � ������ � BetterPolynomial����������� �"mTrue and a number field whose defining polynomial is better, or false if such a polynomial could not be found�g EisensteinIntegerRingX5h���������m EisensteinIntegersR��������P(DT (L����U*/usr/local/magma-2.5/package/RngInt/misc.m8��P�y [Returns the solutions of |f(x,y)| = a, where f is the homogeneous form of the Thue object T�zB�����������.�Input: a finitely generated free group F and two finitely generated subgroups H1 and H2; Output: the subgroup of F which is the intersection of H1 and H2 � � �""""��������@4 � __sig__1Ph� <������  FGIntersect�(���������8� __sig__2t��  MultiRankt �*/usr/local/magma-2.5/package/FldNum/conv.m�������*/usr/local/magma-2.5/package/FldNum/conv.m������� TThe number field L corresponding to the quadratic field K, and the map taking K to L � �@������������� ������(� UThe number field L corresponding to the cyclotomic field K, and the map taking K to L<� __sig__3�� �L�����"gTrue and an order whose defining polynomial is better, or false if such a polynomial could not be found�If�3��������T� __sig__4��'7� �\������HReturns a list with all possible fields L such that L = K1 K2 might hold4�����������p�d� __sig__5� �l������  MergeFields����������h� __sig__0s!The nearest Gaussian integer to xtF����������X�u __sig__5v X�����w0/usr/local/magma-2.5/package/FldNum/anf_compat.m�������x 0/usr/local/magma-2.5/package/FldNum/anf_compat.m�h������� ���������� __sig__9{ The base 10 representation of n|���������X�} __sig__7~ X������ __sig__0���,The number whose base 10 representation is Q����������X�� X�����,/usr/local/magma-2.5/package/FldNum/isprim.m������ True iff a generates the field5���������������� __sig__0��6� �������d5� True iff a generates the field�d5�A������������� __sig__1�d5� ��������ETrue iff element a of order O generates the field of fractions over Q������������� d5� __sig__20� �������-/usr/local/magma-2.5/package/FldNum/subflat.m�����-/usr/local/magma-2.5/package/FldNum/subflat.m�����6The intersection of the subfield lattice elements of Q6��  ������������� __sig__0 z2� ��������63�*/usr/local/magma-2.5/package/Cop/univmap.m ������*/usr/local/magma-2.5/package/Cop/univmap.mh������(The universal map from C to S given by Q����e����������H� __sig__0�� ������  UniversalMap<R'4����������-/usr/local/magma-2.5/package/AlgFP/groebner.m�����-/usr/local/magma-2.5/package/AlgFP/groebner.m�����"""����������( � __sig__0�� �����  PerfectGrouppackage���������p�oReturn the perfect group in the database which is the nth extension of a group of order p^exp by the group baseD \ t �"������h�� __sig__1�� h�����(�MReturn the simple or quasisimple perfect group in the database with name name�� CosetGraphIntersect����������� ������������� __sig__1� �������,/usr/local/magma-2.5/package/FldNum/isprim.m��P��"�����h�� __sig__2� h������ __sig__4sentation� ^Represent the group G by forming a subdirect product of the coset actions on the subgroup in H5��"" �������h��� __sig__3d5� h�����d51 Compute largest possible soluble quotient of F. X�"��P (  �  __sig__1 � �   ����  __sig__6���44/usr/local/magma-2.5/package/GrpFP/solquot/sq_proc.m������4/usr/local/magma-2.5/package/GrpFP/solquot/sq_proc.m�����  SplitExtqr ModulCandidates__si� eThe set of primes p for which a non-trivial p-extension by Q exists in the database of perfect groups ����������h�� h������'O��������� � < � �( __sig__1 X p ) < � ����� *KInitialize a SQProc for a finitely presented group F with expected order n.| � +"O��������� � < � , __sig__2�(- < � ����x.JInitialize a SQProc for a given quotient epi:F -> G with expected order n.� � PerfectGroupExtensions����������� UThe set of tuples <n1, n2> such that Q#p(n1, n2) is is the database of perfect groups�>2����������h�Series�l�Input: a finitely generated free group F and two finitely generated subgroups H1 and H2; Output: a coset graph CG for the intersection of H1 and H2. This graph is the component of the graph CG1xCG2 which includes the vertex (1,1) (where CG1, CG2 are the coset graphs of H1, H2 respectively). The vertex labels of CG are minimal with respect to the Shortlex ordering with regard to the generators of F ���������P�4/usr/local/magma-2.5/package/GrpFP/perfgps/perfgps.m��@���4/usr/local/magma-2.5/package/GrpFP/perfgps/perfgps.m � �oReturn the perfect group in the database which is the nth extension of a group of order p^exp by the group base@��"�����xhl�� __sig__0<\�ht���� ,D�0�Input: a finitely generated free group F and two finitely generated subgroups H1 and H2; Output: the expression computed for the generalized Hanna Neumann conjecture �"""����������XL� T����� h����� � PerfectGroupClassP���������//usr/local/magma-2.5/package/GrpFP/solquot/sq.m���//usr/local/magma-2.5/package/GrpFP/solquot/sq.m���.� Compute a soluble quotient of F. P is a set of primes and the algorithm calculates the largest quotient such that the order has prime divisors only in P. "   ����   8 ������PerfectGroupRepresentationx������������?The set of base groups stored in the database of perfect groups� � � ����������h�h� h������� PerfectGroupBasesHt����������� __sig__6sions<� __sig__5,D\| ,Algebraic relations for the invariant ring R ���������H(0(<(4 0(D(����L 9The ideal of algebraic relations for the invariant ring R� ����������X(0(L(� __sig__1� 0(T(����� )/usr/local/magma-2.5/package/FldFun/pfd.m����� )/usr/local/magma-2.5/package/FldFun/pfd.mX�����' LCM of elements of Qmposition@ =The (complete) squarefree partial fraction decomposition of fl Q��������p((d(H  __sig__0 l ! __sig__1 �<  AddPrimes��=��������� >Emp is a set of primes which are added to the relevant primes for SQP.( ?O���������� < � � @ __sig__6��A < �����B;Replace the set of relevant primes in SQP by the given set.�CO���������� <  �/O��������� < � 0 __sig__3�<R'41 < � ����2-Calculate the set of relevant primes for SQP.3O������������ < � 14 __sig__4D5 < � ����7��������� 82Add the given prime to the set of relevant primes.59O����������� � < � ��1: __sig__5d; < � ����k�"O������� � < � �l  __sig__14X p m < � ����� n(�Start the soluble quotient algorithm for a finitely presented group F with relevant primes restricted by the sequence of tuples f.�o"O������� � < � �p  __sig__15��q < � �����t  __sig__16��~  __sig__18d | "��4  0  __sig__7���  Collectpqrst   NonsplitExt<p, e>,  nopqrstuvw "gInitialize a SQProc for a finitely presented group F without any information about the relevant primes.!"O��������� � < � " __sig__0���"j Compute a soluble quotient of F with order n. If n is 0, then compute largest possible soluble quotient. "�� X  L  __sig__5  T ����"��,   H Compute a soluble quotient of F. S is a sequence of tuples <p, e>, with p a prime or 0 and e a non-negative integer. The order of the quotient will be a divisor of &* [ p^e : <p, e> in S] if all p's and e's are positive. See handbook for further details. e +/usr/local/magma-2.5/package/AlgChtr/chtr.m��p�f :�Remove occurrences of the irreducibles in Irr from the characters in Chs and look for new irreducibles in the reduced set. Return a sequence of new irreducibles found and the sequence of reduced charactersg �J�J�������%�%�%i �%�%������ The n-th odd graph O_n� a�������� &&&� � __sig__0 � � � & &����D � OddGraph 4 � ��������&�  The triangular graph T_n (n > 1),�� a��������8&&,&� __sig__10� &4&����� � TriangularGraphl � ��������0&�  The n-th square lattice graph�"� __sig__2#�#�#� &L&�����%�%�%t ���������%�%�%v �%�%������  ClebschGraph�� ��������x&� The Shrikhande graph � � a���������&&�&� � __sig__5 | � � &�&������ ShrikhandeGraph�� ���������&� &The sequence of the three Chang graphs� ���������&&�&L � __sig__6 � � &�&����<!�  ChangGraphs4"\"� ���������&� The Gewirtz graph�%�%�%(&w CharacterDegrees@ x ���������%X IntersectKernels� Y��������T Z6�Compose the lifts SQ1 and SQ2 of SQP to a new bigger quotient. If the optional parameter Check is set to true, it will be tested whether the intersection of the kernels has maximal index.Ph[OOOO������t < h �\  __sig__11��] < p �����^ ComposeQuotients,Dx  __sig__17cess 0 H N6�Checks whether two soluble quotients have the same kernel. If not and Construct is true, then a bigger quotient will be constructed, where the kernel is the intersection of both kernels.OOOO������D < 8 Q < @ ����R EquivalentQuotientsS��������< T6�Construct a bigger soluble quotient by intersecting the kernels of the given quotient. The return values are a new soluble quotient process and maps from the new to the given soluble groups.UOOO����\ < P V  __sig__10W < X ����d SolubleQuotientProcess e��������� frStart the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being n.� � g"O������� � < � @h  __sig__13�i < � ����Tj,�Start the soluble quotient algorithm for a finitely presented group F with expected order of the quotient being the factorized integer f.0 H  x _��������l &xStart the soluble quotient algorithm for a finitely presented group F without any information about the relevant primes.a"O������� � < � b  __sig__12c < � �����  AppendModule����������� �  internal T l �O������� < � p �  __sig__27H  � < � ������  SolutionSpace����������� �O��������� � < � ��  __sig__28���< � ����� 0 h � SplitCollector ��������� �,�Setup the collector for a standard ssplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series. �O����������L D < 8 ��  __sig__22��� < @ �����O����������T < H ��  __sig__23 x � � rsStart the soluble quotient algorithm for a finitely presented group F with relevant primes restricted by the set s.s"O������� < � u < � ����v"kSet up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p.wO����������� � < � y < � ����z NonsplitCollector{��������� |,�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails.}O���������� � < �  < � �����6�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series.�O���������� < �  __sig__19� <  �����B�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. The pairs in tr indicates trivial tails. ws defines the weights of a series. epi is an epimorphism onto another soluble group.�O���������� <  �  __sig__20� <  �����qSet up the collector for a standard split extension of SQP with algebra RG, R the prime ring in characteristic p.�O����������< , < �  __sig__21� < ( �����4� Calculates the split extension lift for the module M. The return is -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0). d � SplitExtensionSpace� � ���������� �:� Calculates the split extension lift for the modules in the list. Returned is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).0�  __sig__29� � < � ����l �8�Set up the collector for a nonsplit extension of SQP with algebra RG, R the prime ring in characteristic p. ws defines the weights of a series. epi is an epimorphism onto another soluble group.� < P �����6�Calculates the G-modules in characteristic p with respect to the given options. The return value is 0 iff no such module exists, otherwise it is the index of the list of modules in SQP.�O��������t d < X �  __sig__24� <  �������������\ �7Calculates the modules for all (known) relevant primes.�O��������| < p �  __sig__25� < x �����image��T4Append/Find the module M in the list of modules in SQP. The first return value is the index of the module in the char p list in SQP. The second return value is false iff the index belongs to an isomophic module. (It might happen that the isomorphism is the identity; relevant is the internal data structure.)�O������� < � �  __sig__26� < � �����DeleteSplitSolutionspaceH���������d � XDelete the k-th split solution space of the i-th p-module as the actual solution space. ( @ �O����������� < x ��  __sig__39��� < � ������DeleteNonsplitSolutionspace,t���������| �OO��������� � < �  ��O��������� � < � ��>� Calculates the split extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general,otherwise it is the Fq-dimension of the space (could be 0).(@�O��������� � < � t�  __sig__30�� < � ����,�  __sig__31���< � ����h � � �&zBuild the split extension of the i-th p-module for the k-th solution space. A set of linear combinations can be specified.� � � < � ����, � LiftSplitExtension\ t ���������� � XBuild the split extension of the l-th list of p-modules for their actual solution space.���OO������� � < � ��  __sig__41�� < � ����| �IsNodePl�<� Calculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������� � < � �@� Calculates the split extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O�������� < � �  __sig__32� < �����  __sig__40tionspace�6� Calculates the nonsplit extension lift for the module M. The return is -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������  <  �  __sig__33� <  ����� NonsplitExtensionSpace��������� �<�Calculates the nonsplit extension lift for the modules in the list. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).��O��������4 , < �  __sig__34� < ( �����>�Calculates the nonsplit extension lift for the modules in the l-th list in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������D < < 0 �  __sig__35� < 8 �����<� Calculates the split extension lift for the p-modules stored in SQP. The return is a sequence of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������T L < @ �  __sig__36� < H �����@� Calculates the nonsplit extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�O��������\ < P �  __sig__37� < X ����� XDelete the k-th split solution space of the i-th p-module as the actual solution space. �O����������l <  �  __sig__38� < h ���� LiftNonsplitExtension  �������� &  __sig__54ionRow  \Build the non-split extension of the l-th list of p-modules for their actual solution space. OO������< 4 < ( �   __sig__48�(  < 0 ����t WBuild the non-split extension of the list of p-modules for their actual solution space.|��  __sig__49 x  < @ ����� � SBuild the split extension of the list of p-modules for their actual solution space.���OO������� � < � � �  __sig__42� � � < � ����� � QBuild the split extension of the list of modules for their actual solution space.Ht�OO������� < � <�  __sig__43,D� < � ����H OO������L D < 8 d UBuild the non-split extension of the list of modules for their actual solution space.< OO������T < H    __sig__50� �  < P ����D"iBuild the non-split extensions of SQP for the p-modules in the l-th list for their actual solution space.��OO������t d < X �   __sig__514 l �flexes�� eBuild the split extensions of SQP for the p-modules in the l-th list for their actual solution space.�OO������� � < � �  __sig__44� < � ����� LiftSplitExtensionRow���������� � \Build the split extensions of SQP for the list of p-modules for their actual solution space.�OO������ � < � �  __sig__45� < � ���� QBuild the split extension of the list of modules for their actual solution space.OO������ <   __sig__46 <  ����&~Build the non-split extension of the i-th p-module for the k-th solution space. A set of linear combinations can be specified.OO��������, <    __sig__47 <  ����8  PrintSeries��9��������� : Print functionl ;O����������� < � p <  __sig__58H  = < � �����>  PrintPrimes��?��������� @ Print functionAO���������� ! !< !�B  __sig__59 x C < !����� (  PrintProcess )��������� *"General print function for SQProc. � +O����������� < � 0 ,  __sig__550- < � �����. Print function�/O����������� < � \0  __sig__56,1 < � ����� 2  PrintQuotienth 4 Print functionh  <  ����LiftNonsplitExtensionRow��������\  Build the non-split extensions of SQP for the list of p-modules for their actual solution space. OO������� | < p   __sig__52  < x ����  UBuild the non-split extension of the list of modules for their actual solution space.!OO������� < � "  __sig__53# < � ����<Print function for SQProc. Print prime specific information.%O����������� � < � ' < � ����P Print function�QO����������\!L!< @!p R  __sig__62� � S < H!���� T  PrintModules  U��������D!V Print function�WO����������d!< X!X  __sig__63x�Y < !����HZ Print function \  __sig__64� � ^ PrintExtensions� 3��������� 5O����������� < � 6  __sig__577 < � ����[O�����������!t!< h!�] < p!����� _��������l! Print function� aO�����������!< �!@b  __sig__65�c < �!����Td7Returns the soluble group and the epimorphism from SQP.<l�eO�������!< �!h f  __sig__66 h D PrintCollector E��������!F Print function� GO����������!< !� H  __sig__60x � I < !�����J Print function�KO����������4!< (!tL  __sig__61�<M < 0!����|N  PrintRelat0 H O��������,!� < ("����� DeleteSplitCollectorT���������L"�+Delete all collectors for split extensions.� �O����������l"< "��  __sig__75��� < h"����x�BDelete the collector for non-split extensions in characteristic p.x��O�����������"|"< p"| �  __sig__76t � � < x"����|!h  GetQuotient��i���������!j PReturn the relevant primes stored in SQP and a flag indicating the completeness. x � kO�������!< �!0l  __sig__67(@m < �!�����n  GetPrimes<To���������!pGReturns the parent of SQP, i.e. the soluble Quotient which lifts to SQP 0 H r  __sig__68�  s< �!����8!P!x!g < �!����t  GetParent��u���������!v;Returns a list of soluble quotients which are lifts of SQP.� � wO����������!< �!� x  __sig__69��y < �!�����z  GetChildrenHt{���������!|9Returns the i-th soluble quotient which is a lift of SQP.\|}O����������!< �!h ~  __sig__70�!�!qOO���������!< �!� GetChild������������!�'Return the i-th p-module stored in SQP.8  � �Oz��������"< "0 �  __sig__710� < "������  GetModule(���������� "�/Return the list of all p-modules stored in SQP.�,�O���������<","< " �  __sig__72� � <AmbientSys < �!����� DeleteNonsplitCollector ���������t"�/Delete all collectors for non-split extensions.� � � �O�����������"< �"� �  __sig__77��� < �"������*Delete the collectors in characteristic p.l��O�����������"�"< �" � < �"���� � DeleteCollector� ����������"[Finiteants�  GetModules� ���������"�0Return the l-th list of p-modules stored in SQP. � � �O���������D"< 8"� �  __sig__73��� < @"������  __sig__78ectort�>Delete the collector for split extensions in characteristic p.\�O����������d"T"< H"H �  __sig__74�  � < P"����8!� KeepSplitAbelian� ����������#�(�Return the sequence useful for the Collector setup. It determines that the presentation of SQP and of SQR SQP are kept unchanged.� �OO���������#< �#��  __sig__91�(� < �#����t�KeepSplitElementaryAbeliant�����������#�MReturn a sequence of pairs [j,i] s.t. the order of both G.j and G.i is not p.� �O���������#< �#�#�  DeleteProcess ����������"�/Delete the Process and all its child processes.� � � �O�����������"< �"� �  __sig__81��� < �"������ DeleteProcessDown������������"�  __sig__83ete��>Delete all soluble quotient processes which are related to SQPx �O�����������"< �"�!�!Delete all the collectors of SQP.�O�����������"< �"�  __sig__79� < �"�����-Delete the soluble quotient process variable.�O�����������"< �"�  __sig__80� < �"����� DeleteProcessComplete����������"�(Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQP is kept unchanged.� �  �OO��������#< #�� < #����x� KeepGeneratorOrder����������� #�*�Return the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators in SQP is kept unchanged.!<!�  __sig__844"\"�  __sig__82� < �"����� KeepGeneratorAction����������#�(�Return the sequence useful for the Collector setup. It determines that the order of the pc generators in SQRSQP is kept unchanged.�  �OO��������D#< 8#��  __sig__85�,� < @#����x� KeepElementary���������<#�,�Return the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators in SQR SQP is kept unchanged."4"\"�OO��������,#< #� < (#�����  KeepAbelian�����������T#�0�Return the sequence useful for the Collector setup. It determines that the conjugation action of the pc generators of SQP on those of SQR SQP is kept unchanged. | � �OO��������t#< h#�  __sig__87�(� < p#����x� KeepGroupAction@���������l#�qReturn the sequence useful for the Collector setup. It determines that the presentation of SQP is kept unchanged." "�  __sig__884#L#�OO��������\#< P#�  __sig__86� < X#�����  KeepSplit������������#�  __sig__92yAbelian�.�Return the sequence useful for the Collector setup. It determines that the order and the conjugation action of the pc generators in SQR SQP is kept unchanged.l�OO���������#< �#��  __sig__89��� < �#������ KeepElementaryAbelianH����������#�OO���������#< �# "�  __sig__904#L#d#|#�OO���������#< �#� < �#����� < �#����  __sig__94�� < ����@  < @�����  __sig__99ection  TDetermine the maxinmal p-elementary abelian module which lifts to a bigger quotient.� OO������dT< H(  __sig__97 < P����@ ElementaryAbelianSection�� ��������L OO������l< �"  __sig__98�#�#�.�Return the sequence useful for the Collector setup. It determines that the presentation of SQP and the conjugation action in SQR SQP are kept unchanged.� < �#����  IsPureOrder�� �������� HDetermine the maximal p-abelian module which lifts to a bigger quotient. H  OO������<4< (�  __sig__95� < 0���� <�Determine the maximal abelian module which lifts to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.h  OO������D< 8X"  __sig__96#x#� KeepPrimePower ����������#�*�Return a sequence of pairs [j,i] correspnding to the weight condition for the p-group. The weights are defined by the sequence of quotients. 0 ���������< �#�  __sig__93Ht < ����� KeepPGroupWeightst� ���������# 5Check the PC presentation of G for the Hall property.�  ���� < #y ����������%�%�%�{ �%�%����� |  ��������&�%�%� } __sig__2 x � ~ �%�%����� &|The sequence [C_0, C_1, ...] where C_i is the number of irreducible characters of G with degree p^i for the finite p-group G�� j��������&�%�%� � __sig__3 0 h � �%&����� � +/usr/local/magma-2.5/package/Graph/graphs.m(��М�� +/usr/local/magma-2.5/package/Graph/graphs.mp��М�  SplitSection ! ��������t" @�Determine the maximal nilpotent group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.�# OO�������< ��  __sig__100<\% < ����� & RDetermine the maximal p-abelian module with a splitting lift to a bigger quotient.� � ' OO��������< �H#(  __sig__101 X @�Determine the maximal elementary abelian module which lifts to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account. < h���� IDetermine the maximal p-group with a splitting lift to a bigger quotient. OO�������|< p < x����4  SplitElementaryAbelianSectiond5 ���������6 B�Determine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.��7 OO�������< �x8  __sig__104�@9 < ������: MDetermine the maximal p-group with a non splitting lift to a bigger quotient.l ; OO������%�< �|#<  __sig__105��) < �����* SplitAbelianSection+ ���������, >�Determine the maximal abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.- OO�������< �.  __sig__102/ < �����= < �����0 ^Determine the maxinmal p-elementary abelian module with a splitting lift to a bigger quotient.1 OO��������< �2  __sig__1033 < �����H NonsplitAbelianSectionI ��������%J B�Determine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.��K OO������4%< (%L  __sig__108x�M < 0%����HZ  __sig__111belianSection N ]Determine the maximal p-elementary abelian module with a splitting lift to a bigger quotient. "Q < @%����|#V  __sig__110����> NonsplitSection? ���������@ B�Determine the maximal elementary abelian group with a splitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.A OO������ %< %B  __sig__106C < %����P  __sig__109ionD VDetermine the maximal p-abelian module with a non splitting lift to a bigger quotient.E OO������,% %< %F  __sig__107G < %����\ pSection��] ��������d%^ P Determine the maximal nilpotent group with a lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account. Steps puts a limit on the weights of the p-groups of the nilpotent group.Lx_ OO�������%< x%@  __sig__1120Ha < �%����L b NilpotentSection t � c ��������|%d +/usr/local/magma-2.5/package/AlgChtr/chtr.m���0��h __sig__0%L%O OO������T%D%< 8%R  NonsplitElementaryAbelianSectionS ��������<%T B�Determine the maximal elementary abelian group with a nonsplitting lift to a bigger quotient. If PrimeCalc equals true, the relevant primes will be calculated first. If ModuleList is 0, only the known modules will be taken into account.U OO������\%< P%W < X%����X CDetermine a lift with a p-group, given by its lower central series.Y OO������l%< %[ < h%����j RemoveIrreducibles�k ���������%l 4�Make the norms of the characters in Chs smaller by computing the differences of appropriate pairs. Return a sequence of new irreducibles found and a sequence of reduced characters 4m �J�J�������%�%�%xn __sig__1�xo �%�%����p ReduceCharactersxq ���������%r +/usr/local/magma-2.5/package/AlgChtr/pgps.m ����s +/usr/local/magma-2.5/package/AlgChtr/pgps.mh����u __sig__0#�# z __sig__1%t%�%& */usr/local/magma-2.5/package/RngPol/misc.m�����( ��������������(�(�(� ) __sig__0 x � * �(�(����� + 3/usr/local/magma-2.5/package/Grp/is_conj_subgroup.m�����, 3/usr/local/magma-2.5/package/Grp/is_conj_subgroup.mX�����- MWhether a conjugate of N is a subgroup of M, and if so, a conjugating element<. ���������������(�(�(� / __sig__0 � � 0 �(�(�����!1 IsConjugateSubgroup##2 ���������(3 +The set of conjugates of H by elements of G�& '  InternalFixSecondarySubmodule� ��������( 1/usr/local/magma-2.5/package/RngInvar/relations.m��0�� 1/usr/local/magma-2.5/package/RngInvar/relations.mh��0�� (l(����� &SquarefreePartialFractionDecomposition ��������h( 2The (complete) partial fraction decomposition of f�� Q���������((|(� " (�(����� #  PartialFractionDecomposition"�" ���������(% */usr/local/magma-2.5/package/RngPol/misc.m���0��� Centressp� ,/usr/local/magma-2.5/package/RngInvar/prim.m������ ,/usr/local/magma-2.5/package/RngInvar/prim.m������ __sig__0variantsHardX5� Internal function�� ����������'�'�'� �'�'�����  InternalPrimaryInvariantsHard� ���������'� +/usr/local/magma-2.5/package/RngInvar/sec.m0������ +/usr/local/magma-2.5/package/RngInvar/sec.mx������ a��������P&&D&�� SquareLatticeGraph� � � ��������H&� BThe Paley graph of n; n must be a prime power congruent to 1 mod 4� � � a��������h&&\&(� __sig__3l�� &d&����� &�&����� :Secondary invariants for invariant ring R using subgroup H( @ � �����������((�'�'D �'�'����d  )Secondary invariants for invariant ring R,D ���������(�'(x __sig__1 �' (����0  relationsrySubmodule�  Internal 4 l  ����������� (�'(\" __sig__2#d#|# �' (����� __sig__0&@&X&p&�&�  PaleyGraphx� ��������&� The Clebsch graph� � � a���������&&t&L#� __sig__4\� &|&����(&@&X&� a���������&&�&�� __sig__7 T l � &�&����� � GDisplay the Burnside matrix corresponding to the lattice of subgroups LT��� �����������H'8'(','d � ,�The i-th Steenrod operation P^i(f) of f (f must be a multivariate polynomial with coefficients in a finite field and i must be a non-negative integer)�� ����������x'h'l'x� h't'������ SteenrodOperation��� ��������p'� *Return whether invariant ring R is modular�!�!� ����������'h'�'�� h'�'����X&�  GewirtzGraph t � ���������&�  The Cayley graph of the group G���� �������������&&�&� __sig__8x�� &�&����H�  CayleyGraph� � ���������&� -The Schreier graph for group G and subgroup H\"� ���������������&&�&�%� __sig__9&�&�&�  SchreierGraph�� ���������&� (/usr/local/magma-2.5/package/Sys/names.m ������ (/usr/local/magma-2.5/package/Sys/names.m 8����� Assign Q to the names of S� � � ��������������'' '�� __sig__0<T� ''������ SysAssignNamesNum��� ��������'� 1/usr/local/magma-2.5/package/SubGrpLat/burnside.m������ 1/usr/local/magma-2.5/package/SubGrpLat/burnside.m8����� __sig__1trix� __sig__0%p%�%� ('4'�����&�  IsModular��� ���������'� &A free resolution of (the module of) R8 � ����������'h'�' � __sig__2 � � � h'�'����@� __sig__0ion� .A minimal free resolution of (the module of) R�� ����������'h'�'|� __sig__3 0 H � h'�'����� �  The homological dimension of R�!� ����������'h'�'�� __sig__4%�%&� h'�'����|'� DisplayBurnsideMatrix� ��������0'� JDisplay the Burnside matrix corresponding to the lattice of subgroups of G� � �  ����������X'P'('D'h� __sig__1�� ('L'����(� JDisplay the Burnside matrix corresponding to the lattice of subgroups of Gt�� ����������'('T'x � __sig__2 p � � ('\'����x!� ,/usr/local/magma-2.5/package/RngInvar/misc.m�����  ,/usr/local/magma-2.5/package/RngInvar/misc.m����� __sig__0&�&�&�& 'O  InternalSmallGroupProcessRestartx�P ��������0)Q x�Returns a small group process. This will iterate through all groups with order in Orders. To extract the current group from a process, use ExtractGroup(). To move to the next group in a process, use NextGroup(). To find out which group the process currently points to, use ExtractLabel(). The user may limit the process to soluble or insoluble groups by setting Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants thereof).�  R � ��������)P))D)x!T )L)�����"U SmallGroupProcess�#V ��������H)Y __sig__3'<'|'�domain{A CommutatorGroup B ���������(C +/usr/local/magma-2.5/package/Grp/smallgps.m0��h��D  +/usr/local/magma-2.5/package/Grp/smallgps.mx��h��M __sig__1Group� E 7Moves the small group process tuple p to its next group���F ���������� )))tG __sig__0�<H ) )����|I InternalNextSmallGroupx J ��������)S __sig__2pProcessRestart"K :Returns the small group process tuple p to its first group��L ����������8)),)t(�(�(4 �������������(�(�(5 __sig__1�6 �(�(����7 +The set of conjugates of H by elements of G�8 �������������p�(�(�(9 __sig__2���1: �(�(����; -/usr/local/magma-2.5/package/Grp/commutator.m�0��< -/usr/local/magma-2.5/package/Grp/commutator.m�0��= 5The commutator group [H, K] = <(h,k): h in H, k in K>> ��������������)�(�(? __sig__0������@ �(�(�����N )4)�����W *�Returns a small group process as described above. This will iterate through all groups (g) with order in Orders which satisfy Predicate(g). 4 X ��� ��������p)h))$$�Z )d)�����[ "hReturns a small group process as described above. This will iterate through all groups (g) with order o.0H\  ���������)x))l)� ] __sig__4!T!|!^ )t)�����"_ *�Returns a small group process as described above. This will iterate through all groups (g) of order o which satisfy Predicate(g).�'�'((x(� SolvableRadical��� DirectSumDecomposition� CartanSubalgebra$�
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__sig__1gebra$� F True if a Lie algebra of characteristic p > 0 is a restricted algebraT� ����������*�*�*�� __sig__0� �*�*����l � IsRestrictedLieAlgebraT*� FngElementv �������������)�))�)�x )�)����� y *�Returns the first group (g) with order in Orders which satisfies the predicate and the search condition specified by Search (see above). 0z �������������)�))�)t{  __sig__10�t| )�)����\} "lReturns the first group (g) of order o which satisfies the search condition specified by Search (see above). � ~ ����������*)�)�"  __sig__11�#�#� )�)����H%� ��������� **)*$)k

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$rReturns the label (s,n) of the small group process tuple p. This is the order and number of the current group of p(@p �������))�)tq __sig__8�r )�)����,s  InternalExtractSmallGroupLabel t ���������)u B�Returns the first group (g) of order o which satisfies Predicate. The user may limit the the search to soluble or insoluble groups by setting the parameter Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants).�( �� ���������))|)a __sig__5b )�)����$w
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cReturns a list of all groups of order o. The user may limit the search to soluble or insoluble groups by setting the parameter Search. Search may take the value "All", "Soluble" or "Insoluble" (or variants thereof). Some orders will produce a very large list of groups -- in such cases a warning will be printed unless the user specifies Warning := false.���

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$As above, but with a list of orders. l � ���������0*(*) *\"�  __sig__13d#|#� )$*�����$� �����������@*8*),*�(g  InternalSmallGroupProcessIsEmpty t � h ���������)i <Returns the current group of the small group process tuple p$\$j �����������))�)�'� ,/usr/local/magma-2.5/package/AlgLie/cartan.m�����  DerivedSeries� LowerCentralSeries�� UpperCentralSeries�d5� LeviSubalgebra}�  NilRadical�H��6�  CorootSystem�� Degrees Integral��  Dimension, DualKillingForm@D  MatrixGroup����  Irreducible�  OverGroup��  MaxRootLength�  RepPerm���   RootInclusion�   Reflections�  ReflectionMatrices�  SimpleCoroots� SimpleMatrices�  RootLengths< RootRestrictionPT  SimpleRoots��  StandardGroup�WAdvancens  SimpleRepH L-/usr/local/magma-2.5/package/GrpCox/Coxeter.m��� __sig__0��  StandardHom�*  StandardRoots\1Type� NonNilpotentElement� � ���������+� //usr/local/magma-2.5/package/AlgLie/semsimalg.m� __sig__1 � �  ,0,���� Z 8��������p-8-d-�\ 8-l-����t] 9The dimension of the n-th homology group of the complex C<\^ 8���������-8-t-H _ __sig__3 �   8-|-����8!a DimensionOfHomology�"�"b ��������x-k __sig__5ogy%�%c FThe sequence of the dimensions of the homology groups of the complex CT) Pushouton�  AdjointMatrix � ��������4+� *Test whether the Lie algebra L is solvable� � � ���������T+�*H+ � __sig__7Ph� �*P+������ +Test whether the Lie algebra L is nilpotent<T� ����������+d+�*X+,� __sig__8��� �*+����h � ,/usr/local/magma-2.5/package/AlgLie/nilrad.m�� �� ,/usr/local/magma-2.5/package/AlgLie/nilrad.m�� �� $ The nilradical of the Lie algebra L%�%�
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$As above, but with a list of orders.� �����������H*)<*�  __sig__15� )D*����� ,/usr/local/magma-2.5/package/AlgLie/attrib.m�P��� __sig__0��d5� l+x+�����  NilRadical�d5� ��������t+� ) The soluble radical of the Lie algebra L�H��6� �����������+l+�+� __sig__1�d5� l+�+������ -/usr/local/magma-2.5/package/AlgLie/nonnilp.m� �� -/usr/local/magma-2.5/package/AlgLie/nonnilp.m� �� 8 True if the element x of the Lie algebra L is nilpotentd5� �����������+�+�+�� __sig__0d5� �+�+���� �5� ,A non-nilpotent element of the Lie algebra L� �����������+�+�+� __sig__1 � �+�+����0 =  Return v^n��> ��������-�,�,p ? __sig__6 � � @ �,�,���� A lll�������� -�,�B __sig__7��C �,-����xD ComponentProduct��E ��������-F ll��������h. -�,l G __sig__8$ l H�,-�����!I"Return the minimal polynomial of v�#�#J����������0-�,$-�&K __sig__9(�(�(L �,,-�����+ ���������+ ,/usr/local/magma-2.5/package/AlgLie/dirdec.m� � ,/usr/local/magma-2.5/package/AlgLie/dirdec.m� � #The matrix of the Killing form of L� �w�������� , ,,  ,,����  KillingMatrix ��������, __sig__0ition�( EA list of ideals of the Lie algebra L such that L is their direct sum� ���������4, ,(,� ��������� +�*+� __sig__3� �*+����� -The lower central series of the Lie algebra L� ��������� +�*+� __sig__4� �*+����� -The upper central series of the Lie algebra L� ���������,+�* +� __sig__5� �*(+����� FThe adjoint matrix corresponding to the element x of the Lie algebra L� �� ��������<+�*0+� __sig__6� �*8+����M -/usr/local/magma-2.5/package/AlgBas/complex.m�%�N B-/usr/local/magma-2.5/package/AlgBas/complex.m�%�O WCreate the complex given by the list L of maps and such that the last term has degree d� � � P �8��������H-8-<-�Q __sig__0(�R 8-D-�����S "hThe homology group in degree n of the complex C as an A-module, together with the associated epimorphism��T 8��������h-X-8-L-� U __sig__1!""V 8-T-����x#W Homology$�$�$X��������P-Y0The sequence of homology groups of the complex C)�)*[
__sig__2,8,X,,���������,�,�/
The basis of B�	0���������,�,�,�1
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2�,�,����32The index positions of the projective modules of B��4���������,�,�,�5
__sig__4\|6�,�,����x7IdempotentPositions��8���������,9Return whether v is a unit�#�#:���������,�,�,�&;
__sig__5(�(�(<�,�,�����+�CMEmology
SemiSimpleType��������l, +/usr/local/magma-2.5/package/AlgBas/stuff.mP)��,�!+/usr/local/magma-2.5/package/AlgBas/stuff.m�)��,�-
__sig__2rators"���������,�,�#
__sig__0�t6$�,�,�����vl2% IdempotentGenerators& ���������,' ���������,�,( __sig__1� W3) �,�,�����* NonIdempotentGenerators+ ���������,. �,�,�����SEAsitions� //usr/local/magma-2.5/package/AlgLie/semsimalg.mh0��  Create a classical Lie algebra �� ������������+�+�+� __sig__0� �+�+����� SimpleLieAlgebra���7� ���������+� */usr/local/magma-2.5/package/AlgLie/levi.mp.�h0�� */usr/local/magma-2.5/package/AlgLie/levi.m�.�h0�� &A Levi subalgebra of the Lie algebra L� ���������+�+�+ )� __sig__0�� �+�+���� DirectSumDecomposition  ��������,, 0/usr/local/magma-2.5/package/AlgLie/rootsystem.m p0��3� 0/usr/local/magma-2.5/package/AlgLie/rootsystem.m �0��3� +The root system of a semisimple Lie algebra�� ���H9T,D,H,T D,P,�����  RootSysteml� ��������L, */usr/local/magma-2.5/package/AlgLie/type.mX2��3� */usr/local/magma-2.5/package/AlgLie/type.m�2��3� $The type of a semisimple Lie algebra#$ ���������t,d,h,|' __sig__0)<)T) d,p,���� , HasLeviSubalgebra)$)__sig__0*@+�+�+�+u9The quotient complex of C by the sequence S of submodules��v�88���������/�-8-�-�
x8-�-����@yThe direct sum of C and Dx
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__sig__8�$| 8-�-����t} <Create the chain map from C to D all of whose terms are zero|~ 889��������.8-�-h  __sig__9!�!�!� 8-�-����#�  ZeroChainMap#$����������-�The module maps of f'�'�9���������.8-.�,d8���������-8-�-�e
__sig__4	�	�	f8-�-����(gDimensionsOfHomologyph���������-i"The subcomplex of C generated by S��j�88���������/�-8-�-l8-�-����<m
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Subcomplex,Dn���������-oThe quotient complex of C by D� p888���������-�-8-�-�#q
__sig__6$�$�$r 8-�-�����&s QuotientComplex�'t ���������-w __sig__7,�,�,-\-� 88�������� /8-/�� B�Given a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the quotient map to the homology module from the term of lowest degree in the complex on the right end of the complex.��  __sig__28,�� 8-\/������ RightExactExtension� ��������X/� P&Given a complex C of modules over a basic algebra, return the complex of lenth two greater that is obtained by adding the inclusion from the homology to the term of highest degree in the complex and also appending the quotient map from the homology to the term of highest degree in the complex.�&� 88��������x/8-l/�+�  __sig__29--X ��  __sig__10��� 8-.������  ModuleMaps� � ��������.� The kernel complex of fD \ t � 98��������0.8-$.�
	__sig__11��8-,.����(�The kernel complex of fx���98��������@.8-4.��
	__sig__124l�8-<.����� �The image complex of f"�98��������xJP.8-D.L%�
	__sig__13p&�&�8-L.����x(�%The sum of the two chain maps f and g�+�&./usr/local/magma-2.5/package/AlgBas/algebras.m�P��w���������/�/�d5�
__sig__0��/�/�����
MapToMatrix ����������/�
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__sig__1���/�/����0�Returns true if the module is projective and returns a list of how many projective modules of each type are direct summands of the projective cover of the module.�
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.�Returns true if the module is injective and returns a list of how many injective modules of each type are direct summands of the injective hull of the module.t������X0�/L0D
__sig__7��/T0����&~Returns the pushout of the diagram [ M1 <-- fc1 -- D --- fc2 --> M1 ] as an AModule together with homomorphisms from M1 and M2$ wwww����h0�/\0|' __sig__8)<)T) �/d0���� , ��������0]Currentors �/ 0�������6 Given an sequence X of elements in a module over a basic algebra and a sequence N of nonnegative integers, the function creates the homomorphism from the projective module that is a direct sum of N[i] copies of the ith projective module for the algebra to the module of X with the property that the idempotent e of the projective module maps to X[i]*e. w�������� 0�/0� __sig__4� �/ 0���� ^]The projective cover of M given as the projective module P, the surjective homomorphism of P onto M, the sequences of inclusion and projection homomorphism of P from and to its indecomposable direct summands and the isomorphism type of P in the form of a list of the number of copies of the projective modules of algebra of each type that make up P. 3 w00�/$0
__sig__5��4�/,0�����

ProjectiveCover��������(0������H0�/<0�quotmm�LeftZeroExtension� ����������.�8�Given a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the zero map to the zero module from the term of lowest degree in the complex.��88��������/8-�.$�  __sig__24�� 8-/����<� RightZeroExtension\|� ��������/� P&Given a complex C of modules over a basic algebra, return the complex of lenth two greater that is obtained by adding the zero map to the zero module from the term of lowest degree in the complex and also appending the zero map from the zero module to the term of highest degree in the complex.�)�  __sig__25, ,� 8- /�����-. .� 999��������dn.8-T.�  __sig__14� 8-\.����� /The product of the scalar s and the chain map f� 99��������x.p.8-d.�  __sig__15� 8-l.����� -The composition of the two chain maps f and g� 999���������E�.8-t.�  __sig__16� 8-|.����� 0True iff the chain map f is zero in every degree� 9���������.8-�.�  __sig__17� 8-�.����� 8True iff the chain map f is an injection in every degree� 9��������P0�.8-�.�  __sig__18� 8-�.����� 8True iff the chain map f is a surjection in every degree� 9���������H�.8-�.�  __sig__19� 8-�.����� :True iff the chain map f is an isomorphism in every degree� 9���������H�.8-�.�  __sig__20� 8-�.����� *�The dual of the complex C as a complex over the opposite algebra of the algebra of C. The last term of the dual complex is in degree 0.� 88���������.�.8-�.�  __sig__21� 8-�.����� *�The dual of the complex C as a complex over the opposite algebra of the algebra of C. The last term of the dual complex is in degree n.� 88���������3�.8-�.�  __sig__22� 8-�.����� 8�Given a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the zero map from the zero module to the term of highest degree in the complex.� 88���������.8-�.�  __sig__23� 8-�.����� 8-t/���� � ExactExtension�� ��������p/�  __sig__31gy � � 9u���������/8-� �  __sig__30��� 8-�/������ InducedMapOnHomology�� ���������/� NThe subcomplex generated by L. The list L may be either a list of sequences of element of the terms of C or a list of submodules of the terms of C. There must be one element of L for each term of C. The function returns both the subcomplex and the inclusion of the subcomplex into C.�#$��889�������/8-�/|'�8-�/�����)��889�������/8-�//$/L/�  ZeroExtension � ��������/� -Returns the list of the terms of the complex.� � 88��������8/8-,/ �  __sig__26Ph� 8-4/������ @�Given a complex C of modules over a basic algebra, return the complex of lenth one greater that is obtained by adding the inclusion map from the homology to the term of highest degree in the complex to the left end of the complex.� � 88��������H/8-</"�  __sig__270#H#� 8-D/�����$�LeftExactExtension<&T&���������@/�88��������/8-T/�-<�/�0����$=  DimensionsOfProjectiveModules�> ���������0? QThe sequence of the dimensions of the injective modules of the basic algebra$A$. 4 @ ��������1�/�0�A  __sig__15LxB �/1�����C  DimensionsOfInjectiveModules@D ��������1E V7True if the module M is a semisimple module and false otherwise. If true, then the function also returns a list of the ranks of the primitive idempotents of the algebra. This is also a list of the multiplicities of the simple modules of the algebra as composition factors in a composition series for the module.x(�(�(F ������ 1�/1�-H �/ 1�����/� JacobsonRadicalAlgBas � ���������/� >�Given an element x in a module over a basic algebra, creates the homomorphism from the nth projective module for the algebra to the module with the property that the idempotent e of the projective module maps to x*e(@� w��������0�/�/�/t� �/�/����\� LiftHomomorphismt  ���������/ N Given a sequence S = [s_1,s_2, ... ], the function returns a projective module which is the direct sum of s_1 copies of the first projective of the algebra A, s_2 copies of the second, etc. It also returns the sequences of inclusions and projections onto the incecomposable projecitves.�( ����0�/0�- __sig__3/d/|/�/�/� P The quotient of C by the subcomplex generated by L. The list L may be either a list of sequences of element of the terms of C or a list of submodules of the terms of C. There must be one element of L for each term of C. The function returns both the quotient complex and the quotient map.�  __sig__32� 8-�/����� ./usr/local/magma-2.5/package/AlgBas/algebras.mP��� ��Given a module M over a basic algebra and a natural number n the function computes a projective resolution for M out to n steps. The function returns the resolution in compact form together with the augmentation map (P_0 -> M). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (P_{i+1} -> P_i) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable projective modules making up P_{i+1} in the indecomposable projective modules making up P_i and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in P_{i+1} and in P_i.4#L#� ��������82�1,2�&� __sig__2'�'�'� �142����*� CompactProjectiveResolution�,�,� ��������02� __sig__30�01� ZeroMapon� __sig__0Ld� �12������ SimpleHomologyDimensions � � � ��������2� ,�The sequence of sequences of dimensions of the cohomology groups Ext^j(Si,M) for simple modules Si, to the extent that they have been computed.��� �������� 2�12� __sig__1��� �1 2���� � SimpleCohomologyDimensions� � � ��������2� ��������H2� F�The complex giving the minimal projective resolution of M together with the augmentation homomorphism from the projective cover of M into M. Note that homomorphisms go from left to right so that the cokernel of the last homomorphism in the complex is M. .� __sig__4/�/0� �1d2�����1�1�1o  CohomologyRightModuleGeneratorsp ���������1q R)Given projective resolutions P and Q for simple module S and T over a basic algebra A and the cohomology generators for T associated to the resolution Q, the function returns the chain maps of the minimal generators for the cohomology Ext*(S,T) as a left module over the cohomology ring Ext*(S,S).�(r ���������1x1�1s __sig__3+t x1�1�����+u  CohomologyLeftModuleGenerators,v ���������1w IReturns the degrees of the chain maps of the generators of the cohomologyy __sig__4! Pullback��" ��������x0#$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical.Xp$���������0�/�0%  __sig__10�& �/�0����$'IrreducibleModule,t(���������0)$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical.� � * ���������0�/�0�#+  __sig__11 %H%, �/�0�����&-  SimpleModule(t(. ���������05  __sig__13ion\-1  __sig__12$/L/\Dualodule&}Returns the pullback of the diagram [ M1 -- fc1 --> N <-- fc2 -- M2 ] as an AModule together with homomorphisms to M1 and M2.wwww���� K�0�/t0
__sig__9 �/|0����I<Returns the identity (n x n)-matrix over the finite field F.�Jqw��������01�/$1� K  __sig__17�  L �/,1����0 M IdentityMatrix0N ��������(1a __sig__0ismtO @True if the map f is a homomorphism of modules over the algebra.t�P w ��������H1�/<1x Q  __sig__18p � R �/D1����x!S IsModuleHomomorphism"�"T ��������@1U */usr/local/magma-2.5/package/AlgBas/attr.m������V ./usr/local/magma-2.5/package/AlgBas/chainmap.m���W OppositeAlgebraX,x,�,Y CompactProjectiveResolution$/L/g__sig__1erators0�0�0/6�If $v$ is an element of a basic algebra given as a vector in the underlying space, then the function computes the matrix of the action by right multiplication of the element on the algebra.�0���������0�/�0h2�/�0�����3"nTht algebra $A$ as a right module over itself. The module is the direct sum of the projectives modules of $A$.\4���������0�/�0H6�/�0�����7RightRegularModuleP!x!8���������0G	__sig__16iveModules�$9 RThe sequence of the dimensions of the projective modules of the basic algebra$A$.�(�(: ���������0�/�0�-;  __sig__14d/|/Z ���k R*Given projective resolutions P and Q for simple module S and T over a basic algebra A and the cohomology generators for T associated to the resolution Q, the function returns the chain maps of the minimal generators for the cohomology Ext*(S,T) as a right module over the cohomology ring Ext*(T,T).Phl ���������1x1�1�n x1�1����x ���������1x1�1�z x1�1����� {  DegreesOfCohomologyGenerators� | ���������1} -/usr/local/magma-2.5/package/AlgBas/compact.m�x��~ -/usr/local/magma-2.5/package/AlgBas/compact.m�x�� MThe sequence of sequences of dimensions of the homology groups been computed. ,� ��������2�1�1l0 Rank1L1[ CompactInjectiveResolutionx�^  ./usr/local/magma-2.5/package/AlgBas/chainmap.mؖ�_ 2�Given a projective resolutions P for a simple module S over a basic algebra A, the function returns the chain maps of the minimal generators for the cohomology Ext*(S,S).�� ���������1x1|1$bx1�1����t��yGiven a module M over a basic algebra and a natural number n the function computes an injective resolution for M out to n steps. The function returns the resolution in compact form together with the coaugmentation map (M -> I_0). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (I_i -> I_{i-1}) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable injective modules making up I_i in the indecomposable injective modules making up I_{i-1} and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in I_i and in I_{i-1}. The actual return of the function is the compact projective resolution of the dual module of M over the opposite algebra of the algebra of M.�'�'���������P2�1D2�,��1L2����P/�CompactInjectiveResolution�01cCohomologyRingGeneratorsxd���������1e"gGiven the generators for cohomolgy, the function returns the list of degrees of the minimal generators.�"�"�"f���������1x1�1(&hx1�1�����'iDegreesOfGenerators�)�)j���������1�__sig__5ionutionsP/m
__sig__20�0�0����������2�1�2��
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__sig__2bra�$rThe nth injective-syzygy module of M. The cokernel of the nth boundary map in a minimal injective resolution of M.��� ���������2�1�2� __sig__7t�� �1�2����D� InjectiveSyzygyModuleH � ���������2� ./usr/local/magma-2.5/package/AlgBas/creation.mȟ��  ./usr/local/magma-2.5/package/AlgBas/creation.mȟ�� &{Given a finite p-group G and a field k of characteristic p and returns the group algebra kG in the form of a basic algebra.�*�*� q ���������2�2�2�2|/� __sig__00�0�0� �2�2����<2T2l2 NGiven a finite permutation group G, with normal subgroups N and M, such that N < M and M/N is soluble, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of M/N in G/N. If M/N does not have a complement in G/N, the empty sequence is returned� � �  �34����D 5/usr/local/magma-2.5/package/GrpPerm/complement_new.m�p�� 5/usr/local/magma-2.5/package/GrpPerm/complement_new.m�p�� F�Given a finite permutation group G, with normal subgroup M such that M is soluble, if M has a complement in G return a list of representatives of the conjugacy classes of M in G. If M does not have a complement in G, the empty sequence is returned�#�#�# ����������44$4(4 '

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__sig__1�(��2�2����x�~�Creates a basic algebra from a minimal set of relations. The input is a free algebra, the number of idempotents, a list of the right and left idempotents corresponding to the nonidempotent generators of the algebra and a list of relations. The function creates the extra relations that are natural to the idempotents and the multiplications of the idempotents on the nonidempotent generators. Thus the user needs only supply the relations among the nonidempotent generators.�+�+�:���������2�2�20��2�2�����18
MEANSew2�8w������h2�1\2�ProjectiveResolution���������2�D�The complex giving the minimal injective resolution of M together with the inclusion homomorphism from M into its injective hull. Note that homomorphisms go from left to right so that the kernel of the first homomorphism in the complex is M.""�8w�������2�1t2H%��1|2�����&�InjectiveResolution�(�(���������x2�"gThe nth syzygy module of M. The kernel of the nth boundary map in a minimal projective resolution of M.141L1�
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ZeroModule�����������3�The zero map from M to N	�	�	�w��������03�2$3� � __sig__5 � � � �2,3����� ��������(3� +/usr/local/magma-2.5/package/AlgBas/opalg.m����� +/usr/local/magma-2.5/package/AlgBas/opalg.mH����� __sig__1ule � � >��������L3@38!� __sig__0"0"X"� @3H3�����#� ,�Given a basic algebra A, creates the opposite algebra. This is the algebra with the same set of elements but with multiplication * given by x*y = yx.*� @33�����,� OppositeAlgebra .� ��������\3� __sig__21�1 2OverN�2� MakeBasicAlgebra�Z z2� ���������2� 0Creates the tensor product of two basic algebrasB z2� ��������3�2�2�� __sig__3 z2� �23����If�3� The zero A-module� ��������3�2 3�� __sig__4�� �23����� <�Given a basic algebra A and its opposite algebra O, creates the change of basis matrix B from the vector space of A to the vector space of O, so that if x, y are in A then (xy)B is the same as (yB)(xB) in O. , D � w��������|3@3p3�� @3x3������ BaseChangeMatrix�(� ��������t3� &xReturns the dual of a module M over a basic algebra A as a module over the algebra O which is the opposite algebra of A. � � ���������3�3@3�3�"� __sig__3#�#�#� @3�3�����%� .�Given a homomorphism between two modules over a basic algebra A, returns the dual homomorphism between the dual modules over the opposite algebra O of A.--� ww���������3@3�3P1� __sig__42X2p2 *�Given a finite permutation group G, with normal subgroup M such that M is soluble, if M has a complement in G return true, otherwise false. ����������L4$4@4�
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��������|4$//usr/local/magma-2.5/package/GrpPerm/oldtonew.mH��% //usr/local/magma-2.5/package/GrpPerm/oldtonew.mH��& 9Convert an old-style perm group to a new-style perm group4'  ����������4�4�4) �4�4�����ow5* OldToNew�+ ���������4, 9Convert an new-style perm group to a old-style perm group�- � ���������4�4�4. __sig__1���z1/ �4�4����0 NewToOld~921 ���������42 ,/usr/local/magma-2.5/package/GrpPerm/pcore.m��H��3 ,/usr/local/magma-2.5/package/GrpPerm/pcore.m��H��4 "lA minimal elementary abelian normal subgroup of G that lies in the elementary abelian normal subgroup N of G )5   ���������4�4�46 __sig__0�+7 �4�4����9 ���������4H __sig__3nNormalSubgroup: CAn elementary abelian normal subgroup of G, non-trivial if possible;  ���������4�4�4<3< __sig__1�= �4�4����� ���������3� 1/usr/local/magma-2.5/package/GrpPerm/complement.m������ 1/usr/local/magma-2.5/package/GrpPerm/complement.m������ F�Given a finite permutation group G, with normal subgroup M such that M is soluble, if M has a complement in G return a list of representatives of the conjugacy classes of M in G. If M does not have a complement in G, the empty sequence is returned��   ��������4�3�3�3 � __sig__0 � � � �3�3����� *�Given a finite permutation group G, with normal subgroup M such that M is soluble, if M has a complement in G return true, otherwise false.�%(&   �������� 4�34* __sig__1,<,\, �34���� .    �������� 4�342 __sig__22 383T3l3� PathTreeCyclicModule�� ��������D3� ��������d3@3X3� @3�3���� � -The injective hull of the nth simple A-module�� ���������3�3@3�3� � __sig__5 � � � @3�3����� � InjectiveModule� ���������3� 0�The injective module that is the injective hull of direct sum of S[1] copies of the first simple module for the algebra, S[2] copies of the second simple module, etc. � ���������3@3�3x!� __sig__6"�"�"� @3�3�����#� *�Given a module M over a basic algebra A, returns the injective hull I of M as an A-module and the inclusion homomorphism theta: M --> I.)�)*� www�3@3�3L/� __sig__70�0�0� @3�3���� 2� %As above, but with a list of degrees.�� ������������7t6�7< �  __sig__14L d � t6�7����� � __sig__0scription� GReturn the string description of the n-th transitive group of degree d.��� ���������7t6�7�  __sig__15��� t6�7���� � TransitiveGroupDescription� � � ���������7� 0/usr/local/magma-2.5/package/GrpPerm/Aut/autgp.m#0��0��� 0/usr/local/magma-2.5/package/GrpPerm/Aut/autgp.m&���0��� 2Automorphism group of a finite permutation group G@)X)�  "������7�7�7 .� �7�7����p0� 4/usr/local/magma-2.5/package/GrpPerm/Aut/backtrack.m��0���  CartanMatrix3<4�  InjectiveHull�3 HasComplementNew�  ��������D4 NGiven a finite permutation group G, with normal subgroups N and M, such that N < M and M/N is soluble, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of M/N in G/N. If M/N does not have a complement in G/N, the empty sequence is returned(Ht �����������d4$4X4<

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__sig__6#0#H#:8�8�����$; Print the Dynkin diagram of W�&<;����������9�88�8�+= __sig__7,�,�,> 8�8����L/?  DynkinDiagram�0@���������8A-Print the Dynkin diagram of the Cartan matrix�4C __sig__85 6h6&VV��V���������@[email protected]�@�' __sig__3 X p ( [email protected]�@����� ) TrapezoidalQuadratureD *���������@+The Beta function B(z, w),D,VVV���������@[email protected]�@x. [email protected]�@����/  BetaFunction00���������@V */usr/local/magma-2.5/package/RngLoc/misc.m� �� �1<-/usr/local/magma-2.5/package/AlgGeom/specfile|#5 __sig__02.5/packageW 6*/usr/local/magma-2.5/package/RngLoc/misc.mm �� �  __sig__02.5/package�faces�p�E�E����/sch_map.m23/usr/local/magma-2.5/package/AlgGeom/map/equality.m8 ��!�33/usr/local/magma-2.5/package/AlgGeom/map/equality.m� ��!�4YY���������@�@�@76 �@�@�����97  SameDomain�:�:8���������@& �;�<�����'  RootFunction�(���������<)6�Two sequences of matrices. The first represent the fundamental roots, the second represent the negatives of the fundamental roots for the simple Lie algebra of the given type and rank���*;�����<�;�<x+  __sig__11�@, �;�<�����- LieRoots 4 L .���������</&xReturns the groups B, N and H giving the BN-pair for the Chevalley group of the given type and rank over the given field$�$�$0jjj�����<�;�<�(1
	__sig__12�*�*2�;�<�����,3
BruhatChevalley/4���������<53The degree of the representation with weight lambdaT3l36u;���������<�;�<d77
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__sig__9�7"�;�<����#JGiven a nilpotent matrix mm, return the exponential function t -> exp(t*E)�$ X���������<�;�<%  __sig__10B ����������989�D 8 9����� E Degrees of the basic invariants x � F;�������� 9890G __sig__9(@H 8 9�����I  BasicDegreesTJ��������9K8The simple reflection matrices for a given Cartan matrix� � L ��������898,98!M  __sig__100"X"N 849�����#}  __sig__19��~ 8�9������ LongestElement ���������@:�GThe longest word in W as a reduced expression in the simple reflectionsd | � �;��������:8T:�  __sig__24�(� 8\:����x�  LongestWord�@���������X:� TThe set of indices of generators s such that the length of sw is less than that of w!<!��;��������x:8l:$�
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LeftDescentSet�(���������p:�'The unique long root of greatest height/(/P/�;u���������:8�:p2�
	__sig__26�3�3�8�:����L5�
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	__sig__27@9�9OSimpleReflections�%�%P��������09Q'The root system for the Cartan matrix C�+�+�+R0��������X9P98D90S
	__sig__1181P1T8L9�����2U'The root system for the Coxeter group W�4�45W
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	__sig__21��8:�����SThe length of the permutation w as a reduced expression in the simple reflections##��;��������0:8$:X&�  __sig__22@'�'� 8,:�����)�  CoxeterLength$,���������(:�<The longest element in W as a permutation of the root system1�1�;���������H:8<:5�
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	RootNorms!<!t���������9u4The maximum of the squared lengths of the roots of W%�%v;���������98�9X)w
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MaxRootLength�/z���������9{1The finite Coxeter group of a given type and rank�3<4|;��������:�98�9�8�?The reflection subgroup of W generated by the given reflections�P�;;��������;8�:(�
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��UUZ�������B|B�ApB���AxB����$�"The Nagata automorphism of 3-space��QZ���������B�A�B � �A�B���� � NagataAutomorphism!8!����������B�FThe projectivity of A determined by the matrix of base ring elements M$&�DQZ���������B�A�B*�
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__sig__09�9�9ORangentl�HighestShortRootLd����������:�&~The element in the coset H x which keeps all positive roots of H positive. This is the coset representative of shortest length
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	__sig__29�#�#�8�:�����%�SpecialTransversal�&�&����������:�CThe list of all reflections in W as permutations of the root system. .�;���������:8�:2�
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Reflections$6l6����������:�Kformmts����������:�(The unique short root of greatest height�;u���������:8�:� 8�:�����(�Returns a triple f, A, X where the map f : W -> A gives the action of W on its standard G-set X; (in version 0, f is not returned)� � �  __sig__35 ( � 8d;����p OYY��������LADA�@Q �@@A����R XTrue iff f and g are defined by the same functions and have the same domain and codomaint�S00���������PTA�@HAx T __sig__7 p � U �@PA����x!V./usr/local/magma-2.5/package/AlgGeom/map/aut.mhQ�W./usr/local/magma-2.5/package/AlgGeom/map/aut.mhQ�X,True iff the domain and range of f are equal(t(YY��������lA\AA\-Z __sig__0/$/L/[\AhA�����0\
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DualKillingForm����������;�C The value of the Killing form on the root elements e_alpha, e_beta��;���������;�;�;��
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__sig__4ant � �+The structure constant for [e_alpha,e_beta]�"�"�uu;���������;�;�;$&� __sig__3&�& '� �;�;����T)� LieStructureConstant, ,���������;&~The standard graph automorphism of W as a permutation of the roots and a sequence of signs for the action on Lie root elements�3;�������<�;<t8 �; <����4:GraphAutomorphism<;T;l;�;� StandardAction ���������;�.The Coxeter group W as a real reflection group� �;j�������;8t; �  __sig__36Ph� 8|;������ ReflectionGroup(���������x;�AThe fundamental weights with respect to the basis of simple roots\|�;���������;8�;h �  __sig__37�!�!� 8�;����#� FundamentalWeights$X$����������;�)/usr/local/magma-2.5/package/GrpCox/Lie.m�_��a��")/usr/local/magma-2.5/package/GrpCox/Lie.m��a��8The Killing form of the Cartan algebra with Weyl group W0�0�0�; ���������;�;�;�3� __sig__04�45� �;�;����7�DThe Killing form of the dual of the Cartan algebra with Weyl group W:�:9%True iff f and g have the same domain�:00���������@�@�@< ; __sig__1 L d < �@�@����� =YY�������� A�@�@D> __sig__2�? �@�@����X@  SameCodomainxA���������@B'True iff f and g have the same codomainL d | C00��������A�@A�!D __sig__3"# #E �@A����\$FYY��������0A A�@�'G
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__sig__6;@;X;URegular���������<*�The matrix of ad(e_alpha) acting on X (on the right), where X is an indexed set of positive roots or an indexed set of negative roots for W�0u;��������(<�;<�	
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LiftIsomorphism$|���������=}DReturns the zeta function of E at the prime p in the indeterminate q � ~��|������������>�=�=8! __sig__4"0"X"� �=�=�����#�GReturn the order of E with the base (finite) field extended by degree d�&�&�&�|��������>�=> ,� __sig__5-�-�-� �= >�����/�HThe sequence of elliptic curves corresponding to linear Hecke subspaces.|� �������� >��>�6� __sig__9����9� CremonaReference:�:�:������������ __sig__7=\=t=X��������P=Y%Get the current object of the process�4Z ����������p=8=d=�[ __sig__2��7\ 8=l=������7^��������h=_.The Label of the current object of the process� �����������=8=|=a __sig__3�+��7b 8=�=����c  CurrentLabel��7d���������=e)/usr/local/magma-2.5/package/EC/dummy_c.m�{�|�i __sig__0M./usr/local/magma-2.5/package/Process/process.m��N./usr/local/magma-2.5/package/Process/process.m��O(True if no objects remain in the process | � P ���������bH=8=<=4S)Advance to the next object in the process�T ����������X=8=L=�U __sig__1p�V 8=T=�����f)/usr/local/magma-2.5/package/EC/dummy_c.m~���g&zReturns true iff seq defines a point on the elliptic curve E when the base ring is K. Also returns the point if it exists.�#$h��|~�������=�=�=�=�'j�=�=�����)k
IsRationalPoint$,l���������=mB�For E be an elliptic curve on the base ring R, returns true iff seq defines a point on the elliptic curve E on the base ring is K, where f : R -> K is used to coerce the elements of the sequence into K; also returns the point if it exists�8n��|~�������=�=�=�;o __sig__1<�<�<�<=:���������<; eReturns four sequences of functions indexed by the simple roots: x_a(t), x_[-a](t), n_a(t) and h_a(t)d <���<�;�<� =  __sig__14�> �;�<����? ChevalleyFunctions@X@���������<A _The order of the finite Chevalley group of given type and rank over the finite field of order q� 4 l B��������=�;=\"C  __sig__15d#|#D �; =�����$�0/usr/local/magma-2.5/package/EC/SEA/sea_header.m�����./usr/local/magma-2.5/package/EC/SEA/sea_main.m���./usr/local/magma-2.5/package/EC/SEA/sea_main.m���>The order of E using the methods of Schoof, Elkies, and Atkin.�
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	__sig__165x5J�;$=����t8K FactoredChevalleyOrder4:L�������� =R 8=D=�����<��������[email protected])/usr/local/magma-2.5/package/FldRe/real.m���h�� )/usr/local/magma-2.5/package/FldRe/real.m��h�� WA value y and an error estimate dy, such that p(x) = y where p(xa[i]) = ya[i] for all i0PhV�V�VVV������@[email protected][email protected]� __sig__0�� [email protected]\@����� \Approximation to integral of real function f from a to b, using Romberg's method of order 2k h VV��V��������[email protected][email protected][email protected]X" __sig__1##x#  [email protected][email protected]�����$RombergQuadraturel&�&��������[email protected] bApproximation to integral of real function f from a to b using Simpson's rule with n sub-intervals$/L/ VV��V���������@[email protected]|@l2! __sig__23�3�3" [email protected]�@����H5# SimpsonQuadrature77$���������@-
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BaerSubplane=�=�OptimalModularCompositionSetup���������?�$oGiven setup information for a polynomial g and modulus h, set g_r to be the r-th self-composition of g modulo h 4����������� ?�>?05� __sig__3�� �> ?����� OptimalModularCompositionApply���������?�,/usr/local/magma-2.5/package/EC/SEA/schoof.m�7�,/usr/local/magma-2.5/package/EC/SEA/elkies.m��7�(/usr/local/magma-2.5/package/HC/jac_hc.m.m.m���� __sig__0age/EC/SEA/p �=�=����q5Lift the isogeny m to be defined over the base ring K�RH3r�����������=�=�=s __sig__2>5?3t �=�=������CThe number of curves of conductor N and isogeny class S (e.g. "A").�5���������>�=|>8��4�  __sig__12,� �=�>�����-The trace of the Frobenius endomorphism of E.�|���������>�=�>�1�  __sig__13�� �=�>����� TraceOfFrobenius��7����������>� �?�?����� __sig__1����L��� (/usr/local/magma-2.5/package/EC/SEA.spec�7� /home/was/magma/specRngLoc/�D?P?����SEA/atkin.m�+/usr/local/magma-2.5/package/EC/SEA/atkin.migm� __sig__0�-/usr/local/magma-2.5/package/EC/SEA/lercier.m �-/usr/local/magma-2.5/package/EC/SEA/menezes.ml � Change pt1 to pt1 + pt2. /shanks.m�7/usr/local/magma-2.5/package/EC/SEA/modular_equations.mH���7/usr/local/magma-2.5/package/EC/SEA/modular_equations.mH���5Return the canonical modular equation for the prime l�����������T?D?H? ���������L?�3/usr/local/magma-2.5/package/EC/SEA/supersingular.m���H���3/usr/local/magma-2.5/package/EC/SEA/supersingular.m��H���1Returns true if ordinary, false if supersingular.�%$&�|��������t?d?h?*�
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	__sig__19$(t(� �E�G�����*�  BaseSchemex,�,����������G�[V���������G�G�E�2�  __sig__20�384� �E�G����x5�8The subscheme of the domain of f where it is not defined9�9�9�\V���������U�G�E�G�<�  __sig__21�=�=� �E�G�����?�8The subscheme of the domain of f where it is not definedB@CxC�  __sig__22E<E� �E�G����\G� inner_vertices�� vertices�  all_vertices� lower_vertices���� outer_verticesD�  outer_faces���  all_faces��  inner_faces�� islinear���  lower_faces�Q,True iff p = <a,b> lies on the boundary of N�Rb��������PQ�PDQ T �PLQ����8U  IsBoundaryLV��������HQW"True iff p = <a,b> is a point of NXb��������(]hQ�P\Qx Y __sig__6**Z �PdQ����x*[:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/npoly.m� ��!�\:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/npoly.m ��!�])True iff N was defined using a polynomial@8^b���������QpQtQ�C_ __sig__0C�C pQ|Q���� Fa  HasPolynomial0Fe __sig__1N�N/:True iff f is defined by polynomials of degree at most one��0Z���������H�E�H� 2 �E�H����D hR\���������I�I�I�I�j �I�I�����k QuadraticTransformation(l���������ImIThe quadratic transformation of P in the linearly independent points of S� n�TR\���������I�I�I�I|!o __sig__1"�"�"p �I�I�����#qHThe birational pullback of X under the standard quadratic transformation&' 'rVV���������I�I�I�I\,s __sig__2-�-�-t �I�I����0u bThe birational pullback of X quadratic transformation of P in the linearly independent points of S�3<4v�TVV���������I�I�I�8w __sig__39:8:x �I�I����X;y The base change of S to K by automatic coercion together with the induced map of ambient schemes?\?|?z��SSY�������I�I�I�ID| �I�I�����F�ClosureG���������PG�6The underlying map of polynomial rings if f is regular�Z���������GpG�EdG�  __sig__16� �ElG�����  BasePoints�����������G The functions used to define f� Y��������H�EH�   __sig__24� �  �E H���� DefiningFunctions���������H The domain of f<\YP����������(H�E HH   __sig__25�   �E$H����8!The codomain of f�"�"YP���������8H�E,H�%
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	__sig__23����E�G�����O8Declare f to be generically finite or not according to b��PY����������HI�E<I�
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S$Declare f to have (generic) degree d�TY����������UXI�ELIXU  __sig__420xV �ETI�����W4Declare f to be an isomorphism or not according to b L XY����������hI�E\I�!Y  __sig__43�"�"Z �EdI����$[The identity map A -> A�%�%(&\PY��������xI�ElI*]
	__sig__44<,\,^�EtI���� ._$The constant map A -> B with image p0�0TPPY���������I�E|I<4a  __sig__45,5L5b �E�I����d7c  ConstantMap�9�9d���������Ie2/usr/local/magma-2.5/package/AlgGeom/map/special.m(8��9�f2/usr/local/magma-2.5/package/AlgGeom/map/special.mx8��9�{ __sig__4mationBg:The quadratic automorphism [x,y,...] -> [1/x,1/y,...] of P�F�Fi __sig__0G�G�G ��������H  True if f is known to be finite� � � Y���������H�EtHL#  �E|H�����$! True iff f is generically finite&�&�&"Y���������H�E�H�+#
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__sig__6$�$�$� 0R�R����X&� RestrictedSystem'�'�'����������R�?The subsystem of L of projective plane curves tangent to C at pP/h/�/�XTLL���������V�R0R�R�2� __sig__73�3<4� 0R�R����|5�>/usr/local/magma-2.5/package/AlgGeom/linearsystem/elt_linsys.mL��>/usr/local/magma-2.5/package/AlgGeom/linearsystem/elt_linsys.mL��'True if and only if p is a section of L�=(>�>��L���������R�R�R�B� __sig__0D�D�D� �R�R����G�  IsSectionHH� __sig__1J�JXK� IsEllipticWeierstrass ���������$J�3True iff C is an elliptic curve in Weierstrass form�
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��K�K�����<The restriction of f to the jth affine patch of its codomain$�[Z���������K�K�K�K�� __sig__2��� �K�K����� � RestrictionToPatch h ����������K� dThe restriction of f from the ith affine patch of its domain to the jth affine patch of its codomain&T&�\Z���������K�K�K�*� __sig__3,x,�,� �K�K���� /�;/usr/local/magma-2.5/package/AlgGeom/miscellaneous/matrix.m8i�l��;/usr/local/magma-2.5/package/AlgGeom/miscellaneous/matrix.m�i�l��0A sequence containing all the r by r minors of M7�7t8�D���������K�K�K�;� __sig__0<t<�<� �K�K�����=���������K ZTrue if and only if all entries of M are coercible into S; in that case, also return 'S!M'�E�F __sig__1GtG�G�K L����TJ�JTK9,�True iff the compact convex hulls of the defining points of N and M have the same vertices and were defined in a similar way (see manual for details)� :bb�������P�P�P4 ; __sig__1 4< �P�P�����= CompactlyEqual�>���������P?2�True iff the lower vertices of the compact convex hulls of the defining points of N and M have the same vertices and were defined in a similar way (see manual for details)� � @bb������Q�P�PL#A __sig__2$$\B �PQ����(&C  LowerEqual' 'D��������QE2�True iff the outer vertices of the compact convex hulls of the defining points of N and M have the same vertices and were defined in a similar way (see manual for details)(2@2Fbb������ Q�PQ|5G __sig__37 7d7H �P Q�����9I  OuterEqual�:�:J��������QK,True iff p = <a,b> lies in the interior of N?(?Lb��������8Q�P,Q|CM __sig__4DE@EN �P4Q����GO  IsInterior�HIP��������0QS __sig__5M�M�MN�NC���������TD0The degree of the polynomials used to generate LEL��������h^ U,TUpG ,TU�����H3A sequence of basepoints that were used to define L�IL��������,U,T U�J __sig__9�K ,T(U�����L DefiningBasePointsM��������UN&True iff L is a complete linear systemTOL��������DU,T8U�P  __sig__10Q ,T@U����l Z  __sig__12em4(R'The complete linear system containing Lt*x*SLL��������TU,THU�7T  __sig__11 8U ,TPU����8V CompleteLinearSystem<8W��������LUX The dimension of L�CYL���������\lU,TUF[ ,ThU�����N\ The space of coefficients of L�N^  __sig__13�N_ ,TxU��������������������� __sig__2gon�b���������N�N� __sig__0=� �N�N�����>� HackobjCoerceNwtnPgon�7����������N�Print polygon N at level l<?�b�����������N�N�N� __sig__1�?� �N�N����� HackobjPrintNwtnPgona(�5����������N�3The Newton Polygon of a polynomial in two variables��7��b��������0OO�N�N�@� �NO�����  NewtonPolygon��7���������O� __sig__3%  GradedSubTerm�&���������L'IThe terms of f of weight n with respect to the grading on its parent ringH  (�����������L L�L� ) __sig__5�*  L�L����+  GradedTerm�,���������L- PA sequence containing all the derivatives of order m of all the polynomials in f 0 H .���������L�L L�L�!/ __sig__6"�"�"0  L�L����1 AllDerivatives�%2���������L39A sequence containing all the derivatives of order m of f, ,4����������L L�Ll05 __sig__71�1�16  L�L�����27 ]The #I-th derivative obtained by differentiating with respect to the I[i]-th variable in turn78��������������L L�LT;9 __sig__8<D<\<:  L�L�����=; cThe polynomial f with all i-th variable factors removed; also returns the number of factors removed�B�B<���������L L�L�G= __sig__9I�I�I>  L�L����|L ����������xL LlL�  AffineLinearTerm � � ��������pL! QThe terms of f of weight at most n with respect to the grading on its parent ring��"�����������L L�L# __sig__4�  L�L����<P  L<M����Q*The lowest degree among the monomials of f��R���������PM LDM� S  __sig__13� � T  LLM����� UtThe sequence of least degrees of f with respect to each different weighting given by the columns of the 2x2 matrix M�VD�������M LTMW  __sig__14� � X  L\M����� t����������M L�M�"v  L�M�����#wIsHomogeneouslyGenerated%�%�%x���������My;/usr/local/magma-2.5/package/AlgGeom/miscellaneous/scheme.mh����z;/usr/local/magma-2.5/package/AlgGeom/miscellaneous/scheme.m�����{IThe subscheme of S of points at which the jacobian matrix of S drops rank 383|SS���������M�M�M7~ �M�M�����9 JacobianSubrankScheme�:����������M�4Compute a Groebner basis for the defining ideal of S?|?�S����������N�MND� �M N�����F� \Set the defining equations of S to be a Groebner basis of its defining ideal and return themLPL?  RemoveFactor @���������LA cThe common denominator of the rational functions in F and the normalised sequence after its removald#|#B����������M LM�&C  __sig__10((x(D  L M�����*Y MinimumDegrees�Z��������XM[(The polynomial term of least degree of f  � \����������xM LlM0 ]  __sig__150^  LtM�����_ MinimumDegreeTerm�(��������pMaThe polynomial term of degree d in f�b�����������M L�M� c  __sig__16� � d  L�M����"e HomogeneousTermH#f���������Mo  __sig__18ee&�&g:The sequence of coefficients of monomials in f of degree d�)*h����������M L�ML/i  __sig__17�0�0j  L�M���� 2k CoefficientsOfDegree3h3l���������Mm8The polynomial of R of degree d with coefficients from c9�9�9n��������������M L�M�<p  L�M�����=q PolynomialOfDegree@,@r���������M� __sig__1hemeedDsHTrue iff the current set of generators of I are individually homogeneousH�HIu  __sig__19�KL} __sig__0LM,ME CommonDenominator�-�-F��������MGvThe homogenisation of f in degree d with respect to z if this is possible with the grading on their common parent ring�4H�����������8M(M L M�9I  __sig__11�:�:J  LM�����;K Homogenisation�<L�������� MM"jThe homogenisation of f with respect to z if this is possible with the grading on their common parent ringE<EN�����������@M L4MTJO  __sig__12dL|L� OThe Newton Polygon of a polynomial in one variable defined over a Puiseux Field����b�������� O�NO� � �N O����� !!The Newton gradient of the face FPh"b���������PP�P� P�P����% Gradientt�&���������P'1The set of faces that contain the point p = <a,b>p � (b���������PP�P#) __sig__6#�#* P�P�����%+ FacesContaining�&,���������P-&yTrue iff F = <a,b,c> is a face of N (see handbook for details) in which case the normalised tuple of the face is returned�0�0.b�������PP�P�3/ __sig__74�450 P�P����72���������P3:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nbool.mx��x��4:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nbool.mР�x��5 bTrue iff N and M have the same vertices and were defined in a similar way (see manual for details)�D�D6bb�������S�P�P�P�I7 __sig__0L4LLL8 �P�P�����M�S�������� N�MN�� __sig__2 � � � �M N����( � fSet and return the defining equations of S to be a minimal basis of its defining ideal and return themh�S��������0N�MN�� __sig__3��� �M,N�������Z���������@N<N�Mx � __sig__4 p � � �M8N����x!��[���������PNHN�MX� __sig__5%�%�%� �MDN���� '� The pullback f^*(p)<)T)��\���������NXN�MLN .� __sig__6/�/�/� �MTN�����1� The pullback f^*(I)�2�2��Y����������RhN�M\N�6� __sig__78�89� �MdN�����:�T�����������|NtN�M,=� __sig__8>�>�>� �MpN�����@�Evaluate f at the point p�B�B�T�������������N�MxN�G� __sig__9I�I�I� �M�N����|L�  polynomial1�M�M�M�M�=/usr/local/magma-2.5/package/AlgGeom/linearsystem/vs_linsys.m��  is_elliptic��  divisor_group���� is_hyperelliptic � The affine plane curve f = 0�� divisor_class_groupl ������QW���������WDWW8W�� __sig__0�� W@W������2The union of C and D in their common ambient space�WWW���������WTWWHW4� __sig__1H� WPW����� __sig__3ffPlt�&Create the place at the simple point x�W_������dWWXW4(� __sig__2*h*� WW�����*� HackobjCoerceCrvAffPld1���������\W�ITrue iff the point x lies on S or the sequence of coordinates x lies in S88�W��������|WWpW�C� WxW�����C� HackobjInCrvAffPlF���������tW� __sig__5fPl�N� Print the curve C at level l�N�W�����������WW�WTN �M Print L at level l�O�O __sig__0P�P�P ,TlT����@R�R�R����������R�7True if and only if S is a hypersurface in the system L� � 8 �SL���������R�R�R � �R�R����0�  IsElement(@����������R�NThe unique line in the projective plane P through the points of S if it exists<��TRX��������,SS�R�R� � __sig__2 � � � �R�R�����!� OThe unique conic in the projective plane P through the points of S if it exists�����TRX��������S�RS�(� __sig__3*�*�*� �R S�����,�CThe unique hypersurface of degree d passing through the points of S�0�0��TR�������� S�RS84� __sig__45(5H5� �R S����7�  SchemeThrough�9���������S�TTXX��������<S4S�R�>� __sig__5?@,@� �R0S�����A�'The line through p and q as points of C�DE<E�TTWW���������[DS�R8STJ� __sig__6LdL|L� �R@S�����M\ UniformizingElement/AlgGeom� sign���ʸ� __sig__0Q�Q�Qy  MaxFaceLengthz���������Q{HTrue iff N is the Newton Polygon of f (see handbook for precise meaning)��7|��b�������QpQ�Qa(�5} __sig__5[email protected]~ pQ�Q���� IsNewtonPolygonOf�@����������Q�4True iff the face function along F is not squarefree@��7�b�������� RRpQR� __sig__6@��7� pQ R�����@�  IsDegenerate~IsFrees�b���������OPO�O�� __sig__3 T l � PO�O����� �  LowerVertices ����������O�<All vertices of the compact hull of the defining points of N��b���������OPO�Ot� __sig__4�<� PO�O����|  P,P����x   InnerFacesp � ��������(P4Those faces of N of points with minimal y coordinate#b��������HPP<P|' __sig__2)<)T) PDP���� ,  LowerFaces�-�-��������@P9All faces of the compact hull of the defining points of N�2�2b��������PPTP�6 __sig__38�89 P\P�����: AllFaces;l;�;��������XP&The Newton gradients of the faces of N�? b��������xPPlP�D  __sig__4E�E�F  PtP�����G   Gradients0JTJ ��������pP# __sig__5N�N OOdO� GBNewtonPolygon ���������O�!The Newton Polygon of the curve C � �Wb��������@O8O�N,O0 � __sig__4 0� �N4O������7The polygon which is the convex hull of the points of VTl��b��������HO�N<O � __sig__5� � � �NDO����� �:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nvert.m���h���:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nvert.m���h���-The set of points used in the definition of N�&�b��������OPOTO ,� __sig__0-�-�-� PO\O�����/� DefiningPoints 1���������XO� The vertices of N�4�4�b��������xOPOlO�9� __sig__1:|:�:� POtO�����;�/Those vertices of N that lie nearest the origin�=�=�=�b���������OPO|OB� __sig__2DtD�D� PO�O�����F�  InnerVertices�G����������O�/Those vertices of N having minimal y coordinate�N�N O� ���������z� dReturns an expression in terms of Manin symbols for the element v in the space M of modular symbols. � � ut���������z8s�z� �  __sig__84��� 8s�z������ :�Returns an expression in terms of modular symbols of the element v in the space M of modular symbols. We represent a point in P^1(Q) by a pair [a,b]. Such a pair corresponds to the point a/b in P^1(Q).0 H � ut���������z8s�z�!�  __sig__85�"�"� 4 8s�zXd� ConvToModularSymbol�%�%� ���������z� 2Return the basis of M in terms of modular symbols., ,� t�������� {8s{l0�  __sig__86�1�1� 4 8s{Xl�2� ModularSymbolsBasis�4�4� ��������{� tu��������,{ {8s;�  __sig__87<,<� 8s {����t=� ConvFromManinSymbolX?x?� ��������{� tu��������<{4{8s�F�  __sig__88�GH� 8s0{�����K� �tu��������D{8s8{�O�O�O�  AllVertices� ����������O� bAll vertices of the compact hull of the defining points of N together with the origin of the plane( @ �b���������OPO�O�� __sig__5��� PO�O������  OuterVertices�����������O�CThe two end vertices of the face F = <a,b,c> in anticlockwise order� � �b�������OPO�O"� __sig__6#0#H#� PO�O������#True iff p = <a,b> is a vertex of N�&�&�b ���������OPO�O�+� __sig__7,-\-� PO�O����|/� IsVertex0�0�0���������O:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nface.mx��H��:/usr/local/magma-2.5/package/AlgGeom/newtonpolygon/nface.m���H�� The faces of N�:b��������PP P�= __sig__0?X?x? PP����A��������P &Those faces of N which face the origin,G b��������0PPP|L  __sig__1M�M�M3=/usr/local/magma-2.5/package/AlgGeom/curve/intersection_crv.m�4 True iff p lies on both C and Dl � � 5TSS���������X�X�X� 6 __sig__0 x � 7 �X�X�����8 IsIntersection�9���������X:TWW��������Y�X�X,; __sig__1��< �X�X����h =  IsTransverse h >���������X?,True iff C and D intersect transversely at pX@TXX��������Y�X�X�'A __sig__2)�)�)B �XY����X,CTXX�������� YY�X�0D __sig__31�1�1E �XY����43F8The local intersection number of the curves C and D at p5 6h6GTWW��������,YY�XY�:H __sig__4;�;�;I �X Y�����<J The intersection number (C,D)?KXX��������4Y�X(YxCL __sig__5DE<EM �X0Y����\GOWW��������TYDY�X8Y�LP __sig__6M�M�MQ �X@Y���� PR IntersectionPoints<QTQw __sig__4Px pQ�Q����l ���������R�LTrue iff the defining polynomial of N is nondegenerate an all the faces of N � �b��������(RpQ R(� __sig__7l�� pQR�����>/usr/local/magma-2.5/package/AlgGeom/linearsystem/sub_linsys.mH���6Return a random hypersurface from the linear system L.��LV���������TLS�S�  LST����   RandomElement ���������S6Return a random hypersurface from the linear system L.,LV�������� TLST�  __sig__10x LST����  GeneralElement� ��������T :/usr/local/magma-2.5/package/AlgGeom/linearsystem/linsys.m������ .:/usr/local/magma-2.5/package/AlgGeom/linearsystem/linsys.m�����  ambient�'((x( sections)�)*  is_complete<,\, defining_base_points� ����������� complete_linear_systemp2 �����������L����������pT,TdT�9  HackobjPrintLinSys�:�: ��������hT L���������T,T[email protected]  __sig__1AtA�A! ,T�T����D" HackobjCoerceLinSys�E�F#��������|T+The complete linear system of degree d on P�L�L%RL���������T�T,T�T�P& __sig__2Q�Q�Q' ,T�T�����S�LL���������b�SLS�S� __sig__6����5� LS�S�����,The hypersurface common to all elements of L�LV���������SLS�S��7� __sig__7�%� LS�S����(  LinearSystem�)���������T*HA maximal linear system of degree d on P whose elements do not contain CD+XRL���������T�T,T�T�, __sig__3�- ,T�T�����.>The linear system on P whose sections are the polynomials in F�/��RL���������T,T�T0 __sig__4 1 ,T�T����42>True iff x is a polynomial section of L or a hypersurface of L3L���������T,T�T4 __sig__5  5 ,T�T����x 6 HackobjInLinSys*7���������T85The ambient space containing the schemes comprising L�79LP���������\�T,T�T(8: __sig__68<8; ,T�T����T8=���������T>"The polynomials used to generate LF?L��������U,T�T�N@ __sig__7N�NA ,TU�����NB SectionsTHTF __sig__8T�G coefficient_space4JXJ  base_points����complete_coefficient_space�NO  dimension8PPP  �����������  base_locus�S T�  BaseComponent�7����������S�<Remove any common factors from the generating functions of LL�LL����������SLS�S� __sig__8P� LS�S����,R __sig__9�>/usr/local/magma-2.5/package/AlgGeom/linearsystem/sub_linsys.m����TLL��������HR<R0R��7� __sig__0�=� 0R8R�����  Subsystem�>���������4R��TLL��������XRPR0R� __sig__1�>a��7� 0RLR�����&yThe linear subsystem of L of hypersurfaces vanishing at a point or sequence of points or single point with multiplicity m[�t6�TLL��������hRR0RTR�@� __sig__2��7� 0R\R������7� VThe subsystem of L determined by the subspace V of the complete coefficient space of L��7�tLL��������xRpR0RdR� __sig__3@��7� 0RlR�����@�:The subsystem of L whose sections are the polynomials in F�  ImageSystem������������V�4/usr/local/magma-2.5/package/AlgGeom/curve/aff_crv.m������RX���������k�W�W�W� �W�W����(�2The union of C and D in their common ambient spaced�XXX���������W�W�W�� �W�W����� __sig__3rojPl��&Create the place at the simple point x��X_������X�W�W� __sig__2  �W�W����4 HackobjCoerceCrvProjPlL���������WITrue iff the point x lies on S or the sequence of coordinates x lies in SH X��������X�W X* �WX����x* HackobjInCrvProjPl�*��������X __sig__5ojPl8 Print curve C at level l8,8 X����������0X�WX�C  __sig__4C�C  �W,X�����C  HackobjPrintCrvProjPl F��������(XDAn affine patch of C centred at the point p and the embedding into C�NTXW[������HX�W<XW �WDX����,W CentredAffinePatch�� singular_germs�U� �V�V������ SThe linear system of hypersurfaces of degree d in the range of f which contain f(S)��V\L���������V�V�V� __sig__2�%� �V�V����� 8s�z����� VQ3�     !"#%&'()*+,-./0123456"789:;<=>?@ABCDEFGH�IJKLMNOPQRSTUVWXY25]^_ E#TpDLxF#���������G#"iComputes Tp on the subspace of degree 0 divisors in the Mestre module. p must be a prime between 2 and 7.d | H#8 ��������(�8s ��!I#  __sig__314# #J# 8s�����\K#TpXt%�%L#�������� �c#  __sig__319torX)M#6Computes the Eisenstein factor of the Mestre module X.-N#88��������@�8s4�P1O#  __sig__315X2p2P# 8s<�����<4Q# MestreEisensteinFactor|5R#��������8�S# Returns the monodromy weights.�:T#8��������X�8sL�x=U#  __sig__316?(?V# 8sT������@W#6Returns the basis for D of supersingular j-invariants.�DX#8��������Ѝh�8s\�IY#  __sig__317LLZ# 8sd������M[# SupersingularBasis�O�O\#���������]#NReturns the decomposition of X into Hecke stable submodules under T2,T3,T5,T7.�U_#  __sig__318WlW�W�W� __sig__4<K� W�W����� HackobjPrintCrvAffPl�@����������W�2The number of geometric punctures of C at infinity�@��7�W���������WW�W� W�W�����IsFlexrty�=/usr/local/magma-2.5/package/AlgGeom/linearsystem/vs_linsys.mj��7�LLL��������dS\SLSPS� LSXS����<3�KA complement in L to the space of sections which contain the hypersurface Sڇx6�VLL��������lSLSS�� __sig__1��7� LShS����+��7�FThe linear system containing hypersurfaces belonging to both L1 and L2�7�LLL���������c|SLSpS� __sig__24=� LSxS�����  Intersection���������tS�FThe linear system containing hypersurfaces belonging to both L1 and L2<?�LLL���������c�SLS�S� __sig__3�?� LS�S�����(True iff L1 is a linear subsystem of L2.��LL���������SLS�S� __sig__4�@��7� LS�S�����  IsSubsystem��7����������S�(True iff L1 is a linear subsystem of L2.�@��7�LL���������b�SLS�S� __sig__5@��7� LS�S�����@�KTrue iff L1 and L2 are equal subsystems of a common complete linear system.} ,T�U�����  __sig__83���#����������#�����������8s� �#  __sig__326| � �# 8s �������# SerreModPQTableNM��#�������� ��#  __sig__328Connected�# RReturns True if and only if the subtori intersection graph of J_k(N) is connected.4 L �#��������<�8s0��!�#  __sig__327�"�"�# 8s8����� �# IsIntersectionGraphConnected%(&�#��������4��#�� Given an integer N (the conductor), a sequence [d1,...,dr] of degrees which define a mod N character eps, and a sequence [p1,p2,...,pn] of primes p such that a_p=0, this function computes the subspace ker(Tp1') meet ker(Tp2') ... meet ker(Tpn') of the space M_5(Gamma_1(N),eps;F5bar)^+. (Primes denote transpose.) The first argument returned is the subspace W, and the second is the space of modular symbols. The subspace W is a subspace of the dual and can be computed efficiently using FastTp.<H<<�#tt������T�8sH�[email protected]�# 8sP������A�#��������L��#Artin2List�#����������h�8sL�#  __sig__329M0M�# 8sd�����hO�# TableEigenvalueList�P�P�#����������#  __sig__330�T�T]Lt��������|U,TpU  CoefficientSpace d � a��������tUj  __sig__15tSpace bDThe space of coefficients of the complete linear system containing L�cLt���������U,T�Ud  __sig__14x�e ,T�U����H���������U�FThe coefficient vector of f in the complete linear system containing L� ��Lu�������� V,TV� �  __sig__20� � � ,TV����� CoefficientMap����������V� RThe polynomial on the ambient space of L defined by the coefficient space vector v,D�uL���������V,TV� �  __sig__21P!x!� ,T V�����"�  PolynomialMap�#��������� V�3The generic multiplicity of hypersurfaces of L at p))�TL��������tc<V,T0V�-�  __sig__22�/�/� ,T8V����L1�:/usr/local/magma-2.5/package/AlgGeom/linearsystem/pencil.mx&�(*��:/usr/local/magma-2.5/package/AlgGeom/linearsystem/pencil.m�&�(*�� Print L at level l�9�9�M����������TVDVHV�<� __sig__0=�=�=� DVPV�����?� HackobjPrintPnclApA�A���������LV�M��������hVDVI� __sig__1K�KL� DVdV����|M� HackobjCoercePncl�O�O���������V�%The pencil on P determined by f and gT� __sig__2UXU�U�U�UfCompleteCoefficientSpace � � g���������Uh The base scheme of L"�"iLV���������U,T�U&k ,T�U����|'l The base scheme of L)�)mLV���������U,T�U /n  __sig__1640l0o ,T�U�����1p=The points of the base locus of L if that is zero-dimensional�4qL���������U,T�U�9r  __sig__17|:�:s ,T�U�����;t?True iff the hypersurfaces of L have no common geometric points�>�>�>uL���������U,T�U�Bv  __sig__18�D�Dw ,T�U����,Gx IsBasePointFree�Hy���������Uz?True iff the hypersurfaces of L have no common geometric points�O�O P{L���������U,T�U�S|  __sig__19U0UXU�U���RM���������VDVtV�� DV|V����� �=/usr/local/magma-2.5/package/AlgGeom/linearsystem/maplinsys.m� 8��=/usr/local/magma-2.5/package/AlgGeom/linearsystem/maplinsys.m� 8�� \The map from the ambient space of L to some projective space determined by the sections of LT�L\���������V�V�V�V,� __sig__0��� �V�V����h �sThe map P - -> Q determined by L; P is the ambient space of L, Q is any projective space of the same dimension as L#x#�LRR\���������V�V�V�&� __sig__1'(t(� �V�V�����*�NThe linear system on the domain of f determined by the polynomials defining f.|/�\L���������V�V�V�2� __sig__23�384� �V�V����x5� HomoloidalSystem7�7t8����������V�@/usr/local/magma-2.5/package/AlgGeom/linearsystem/image_linsys.m<X5� 8��@/usr/local/magma-2.5/package/AlgGeom/linearsystem/image_linsys.m@�5� 8��0The subsystem of L of hypersurfaces containing SDtD�D�VLL���������V�V�V�H� __sig__0K�K�K� �V�V����,M�BThe polynomials of degree d in the range of f which vanish on f(S)�P�P�V\���������V�V�V�T� __sig__1U�UV(VXVZ'"nReturns an echelon basis for the subspace of W-anti-invariant theta functions for the Brandt series matrix TM.@ \'  __sig__11d � ]' ��p������ ^' AntiInvariantTheta�  _'��������l�'6Returns an echelon basis for the kernel subspace of W.,a'  ���������������b'  __sig__12xc' �������� d'  ThetaKernel� � e'����������f'*/home/was/magma/Kohel/modular_isogenies.mg�:��@�g'*/home/was/magma/Kohel/modular_isogenies.mg0;��@�h' fReturns the sequence of points on X specifying isogenies of the elliptic curve E0 over its base field.�+i'�|���������������/j' __sig__00�01k' ��������p2l'  ModularPoints�3m'����������n'qReturns the image curve F and the kernel polynomial of the isogeny E -> F corresponding to the modular data X, P.p;�;o'�||o������ĝ�����>p' __sig__1@[email protected][email protected]q' ��������Bz' __sig__3umsD�Dr' _Given the sequence of successive power sums, returns the sequence of successive symmetric sums.�I4JXJs'��������ԝ��ȝOt' __sig__2P8PPPu' ��Н�����Qv' PowerToSymmetricSumsS�Sw'��������̝{' �������8X|'  VeluIsogeny�Y�YgResIntersectionAtInfinity>2h���������Yi;The points of C in the complement of its first affine patchjX���������YdY�Y'l dY�Y����m ResPointsAtInfinity�n���������Yo</usr/local/magma-2.5/package/AlgGeom/curve/singularity_crv.mB�pL�p.</usr/local/magma-2.5/package/AlgGeom/curve/singularity_crv.mB�pL�qW���������Y�Y�Yr __sig__0�+s �Y�Y����t True iff C is reduced;��4uX��������<b�Y�Y�Yv __sig__1L��7w �Y�Y�����xW���������Y�Y�Yy __sig__2���7z �Y�Y����{ The genus of C5|X���������Y�Y�Y�} __sig__3��7~ �Y�Y����+��7W��������ZZ�Y� __sig__4�d5� �YZ�����>� GeometricGenus����������Y�The geometric genus of C8?�X�������� Z�YZ� __sig__5?� �YZ�����?�The arithmetic genus of C�?�X��������,Z�Y Z� __sig__6�A� �Y(Z����� ArithmeticGenus���������Z� The Milnor number of C�W��������DZ�Y8Z�M� __sig__7�@� �Y@Z�����  MilnorNumber@��7���������<Z�,True iff the origin is a singular point of C�@�W��������dc\Z�YPZ� __sig__8�@��7� �YXZ����' __sig__9�' d������'JSubspace of V orthogonal to U with respect to the Euclidean inner product.4'ttt�������� �d���'  __sig__10�' d�������''/home/was/magma/Kohel/modular_curves.mg�S� ''/home/was/magma/Kohel/modular_curves.mg�S� '4Returns the value of f under the pth Hecke operator. '������������4��(�4 ' �0�����L'  HeckeImage'��������,�'"iReturns the Brandt module of weight one theta functions for the binary quadratic forms of discriminant D.| ' ��������L��@��*' __sig__11d1' �H������7' BinaryBrandtTheta8'��������D�'4Returns a basis of weight two cusp forms of level N.\8'��������t�d��X��C' __sig__2F F' ������8F'  ModularBasis�N'��������\� 'GReturns a basis of weight two cusp forms of level N and precision prec.T W '��������|��p��W ' __sig__3_  codimension���ideal�� The projective plane curve f = 0˸� __sig__0� 4/usr/local/magma-2.5/package/AlgGeom/curve/aff_crv.mT�p\�� geometric_genus�1�T�p\�� singular_origin<3� singular_subscheme��7� singular_points�� ConvToManinSymbol� #=Compute the dimension of the new subspace of S_k(Gamma_0(N)).< #��������d�8sX�@ #  __sig__306 0 # 8s�����h# DimensionSknew�#��������\� #=Compute the dimension of the new subspace of S_k(Gamma_0(N)).� #��������|�8sp�x #  __sig__307p � # 8sx�����x! # DimensionSkG1new"�"�"!#��������t�"# dCompute lots of information about dimensions of modular forms at level N and weight between 2 and k.)T)##������������8s�� .#  __sig__308�/�/%# 8s�������1&#  DimensionAll2�2'#����������(#?Returns a supersingular j-invariant in characteristic p < 10^6.9<9�9)#r����������8s��,<*#  __sig__309=,=+# 8s������?,# FindSupersingularJ�@�@-#����������.#r����������8sH/#  __sig__310�JTK0# 8s�������L2#����������:# 8s������]�;#Mestre�Q<#���������=#+Returns the Mestre method of graphs module.�W�W�WX�  Punctures�(����������W�5/usr/local/magma-2.5/package/AlgGeom/curve/proj_crv.m��d�� 5/usr/local/magma-2.5/package/AlgGeom/curve/proj_crv.m��d�� patchesc_genush � jacobian_ideal�!� jacobian_matrix�"##� __sig__1roup��  singularities�%&TXW��������f�XXX�X�( XX�X����� )TWW���������X�XXX� * __sig__4 | � + XX�X�����,  IsTangent��-���������X.1True iff C and D are nonsingular and tangent at p�@/TXX���������XXX�X� 0 __sig__5 � � 1 XX�X�����!2=/usr/local/magma-2.5/package/AlgGeom/curve/intersection_crv.m� h�S��������<YT"The intersection points of C and D�(�(UXX��������\Y�XPY�-V __sig__7/h/�/W �XXY����1X6/usr/local/magma-2.5/package/AlgGeom/curve/resultant.m h�Y6/usr/local/magma-2.5/package/AlgGeom/curve/resultant.m h�ZWW���������YpYdY�:[ __sig__0;�;�;\ dYlY����=] ResIntersection�>^��������hY_ eA sequence containing the intersection points of C and D; an error if C and D have a common componentGXX���������YdY|YPLa __sig__1MhM�Mb dY�Y�����Ok __sig__3InfinitycLThe common points of C and D in the complement of their first affine patchesT�Te __sig__2U�UVfdY�Y����8XPXlX��������@X8/usr/local/magma-2.5/package/AlgGeom/curve/tangent_crv.m,h�m� 8/usr/local/magma-2.5/package/AlgGeom/curve/tangent_crv.m0�h�m�TWW��������tXdXXX�3 __sig__04�45 XXX����7  TangentLine@9�9��������\X 4The tangent line to C at p embedded as a plane curve=0= TXX��������|XXXpX(A  __sig__1B�B�B  XXxX����@E TWW���������X�XXXXJ! __sig__2LhL�L" XX�X�����Mx',�Returns the isogeny E0 -> E1 with kernel defined by psi0, such that the invariant differential on E0 pulls back to the invariant differential on E1. � y'o|������������0 }'���������~' Returns the kth derivative of f.(�'QQ�������� ������\�' __sig__4,�' �������� �' Returns the kth derivative of f. p � �'oo�������� �����#�' __sig__5#�#�' ��������%�'MReturns the derivative with respect to the fixed generator of the base field.�)�'oo��������,����� /�' __sig__6/40l0�' �� ������1�' Returns the derivative of f.3h3�'QQ��������4���(�7�' __sig__79�9�9�' ��0������:�'o|o��������@����=�' __sig__8?X?x?�' ��<�����A�'��������8��'|Q��������T����H�' __sig__9K�K�K�' ��P�����,M�' PhiOmegaN�N O�'��������L��' eReturns codomain E1 of the isogeny of Velu, defined with respect to the monic kernel polynomial psi0.�U�'  __sig__10�W�W�(RSeriesn#  TangentCone4PLP���������X%4The tangent cone to C at p embedded as a plane curveU�U' __sig__3V�VhW�,The line through the distinct points p and q �TTS���������[�[�[� � �[�[����� �8/usr/local/magma-2.5/package/AlgGeom/curve/weierstrass.m 0v��}��8/usr/local/magma-2.5/package/AlgGeom/curve/weierstrass.m�v��}��Flexesst�TXX\|�����[�[�[�� __sig__0\|� �[�[����x � WeierstrassForm� ����������[�T2A projective curve W in Weierstrass form isomorphic to the (closure of the) nonsingular genus 1 curve C. The second return value is a map of the ambient projective space restricting to an isomorphism W -> C taking the flex at infinity to p. The third return value is an expression of W in the CurveEll type�. /�TWW\|�����[�[�[<2� __sig__13P3h3� �[�[����5�>�A curve W in Weierstrass form isomorphic to the nonsingular genus 1 curve C. The second return value is a translation of the ambient space restricting to an isomorphism W -> C taking the flex at infinity to the flex at p=�=�TXX\|�����[�[�[�A� __sig__2CDD\D� �[�[�����F� WeierstrassFormFlex�G�G����������[  polynomial���:�A curve W in Weierstrass form isomorphic to the nonsingular genus 1 curve C. The second return value is a map of the ambient space restricting to an isomorphism W -> C taking the flex at infinity to pU�U�U�TXX\�������[�[�[0Z�Flex��  __sig__18H� �Y[������TW��������4[[�Y �  __sig__19X p � �Y [����� ��������� [�+True iff p is an ordinary double point of C���TX��������<[�Y0[��  __sig__20\|� �Y8[����x �TW��������X[H[�Y�!�  __sig__21##� �YD[����X�IsCusp�%���������@[�-True iff p is a nonordinary double point of C�+�TX��������[�YT[�/�  __sig__22�0 1� �Y$����l2�7/usr/local/magma-2.5/package/AlgGeom/curve/blowup_crv.m����7/usr/local/magma-2.5/package/AlgGeom/curve/blowup_crv.m����8The two natural patches of the blowup of C at the origin;l;�;�WWW�������[x[h[l[�>� __sig__0@[email protected][email protected]� h[t[����B�Blowup\D���������p[�?The birational transform of C under the toric blowup given by M�K�KL�DWW�����[h[�[�O� __sig__1P�P�P� h[�[����@R�7/usr/local/magma-2.5/package/AlgGeom/curve/crv_common.m����7/usr/local/magma-2.5/package/AlgGeom/curve/crv_common.m���� __sig__0ZtZ�Z�  Singularities�7���������hZ�GA sequence of the singular points of C defined over the base field of C�@��7�X�������Z�Y|Z�@�  __sig__10�7� �Y�Z������7�W���������Z�Z�Y�  __sig__11��7� �Y�Z�����@� SingularPoints����������Z�GA sequence of the singular points of C defined over the base field of C��7�  __sig__12�  __sig__13dXX���������YdY�Y�� __sig__3��� �[�[�����;' ����������������� >' YThe theta series with character Chi of the lattice with Gram matrix M, to precision prec.� � ?' ���������������(@' __sig__6l�A' �������B'>The derivative f' - k*E2*f/12 of the modular form of weight k.�C'����������������� D' __sig__7!""E' �������x#F' ModularDerivative��G'�������� �H'2Returns an echelon basis for the Eisenstein space.�*�*I' ��������,��� �|/J' __sig__80�0�0K' ��(�����<2L' EisensteinSpaceh3M'���������N',Returns an echelon basis for the cusp space.9�9O' ��������D���8�,<P' __sig__9<=,=Q' ��@�����?R'  CuspSpace�@�@S'��������<�T'"iReturns an echelon basis for the subspace of W-invariant theta functions for the Brandt series matrix TM.�I�IU'  ��������\���P��NV'  __sig__10�O PW' ��X�����TQX' InvariantTheta�RY'��������T�['  ��������t���h��Y� IsOriginSingular  ( ���������TZ�W�������ZpZ�Y�� __sig__9(�� �YlZ������X���������Z�Y�Z � �Y�Z���� �  __sig__15verExtension �W���������Z�Z�Y�"� �Y�Z����� HasSingularPointsOverExtension&����������Z�LFalse iff all the singularities of C are defined over its current base field-\-�X���������Z�Y�ZL1�  __sig__14T2l2� �Y�Z����84�TW�������Z�Z�Y�8� �Y�Z����d:����������Z�True iff p is a flex of C\=t=�TX�������Z�Y�Z�A�  __sig__16@CxC� �Y�Z����lE�TW��������[[�YTK�  __sig__17�L�L� �Y�Z�����N�  IsDoublePoint P����������Z�!True iff p is a double point of C�T�T�TX��������[�Y [�X/ The ambient space containing S0SP���������\8\�\� 2 8\�\���� 3 The dimension of S� � 4S��������g�\8\�\@5 __sig__7�6 8\�\����T7)The codimension of S in its ambient spacet�8S���������\8\�\x 9 __sig__8 p � : 8\�\����x!;  Codimension�"�"<���������\=2True iff the ambient space of S is an affine space�&�&>S��������]8\�\ ,? __sig__9-�-�-@ 8\�\�����/A IsAffineScheme 1B���������\C5True iff the ambient space of S is a projective spacex5DS��������]8\ ]d:E  __sig__10<;T;F 8\]�����<G IsProjectiveScheme�=�=H��������]I True iff S is a point�BJS���������_0]8\]\GK  __sig__11�HIL 8\,]����LLM*True iff the ambient space of S is a plane�N ONS��������@]8\4]�QO  __sig__12�STP 8\<]�����UQ IsPlanarW�W�WR��������8]U  __sig__13L[|[ WeierstrassFormNonFlex���������[2/usr/local/magma-2.5/package/AlgGeom/curve/duval.m��(��2/usr/local/magma-2.5/package/AlgGeom/curve/duval.m8��(��TX��������(\\ \� __sig__0Ht  \\�����IsAnalyticallyIrreducible�<��������\ %True iff C has exactly one place at p TW��������0\ \\�"  __sig__1#�#�#   \,\�����% 1/usr/local/magma-2.5/package/AlgGeom/scheme/sch.m���(��&1/usr/local/magma-2.5/package/AlgGeom/scheme/sch.m��(�� dimension_procedure. .! __sig__1��% __sig__2������������������������> __sig__1l��)The scheme defined by the function 1 on A�:�:PS��������\8\T\t= __sig__0> ?? 8\\\�����@% __sig__3BHB�B& 8\�\�����D' The base ring of SG,G(S����������g�\8\�\|L) __sig__4M�M�M* 8\�\�����O+0The base ring of S iff S is defined over a fieldQ�Q�Q,S�����������f�\8\�\XV- __sig__5X X4X. 8\�\����0Z1 __sig__6[L[|[�[�[�S���������\^^P^�� ^X^����� ' �x�����( �& __sig__5eld @ '2Returns the function field of the modular curve X.0!'����������������t"' __sig__4�t#' �������\�&����������������H �& ������� %' ��������x!&' EllipticCuspForm"�"�"''����������(''The cusp form of a minimal model for E.�&�&�&)'|���������������� ,+' ���������-,' CuspForm/�/�/-'����������.'@Returns the zeta function at p of the rational elliptic curve E.4�4�4/'|����������Ĝ�����90' __sig__2:�:�:1' ���������;<' __sig__5ynomial<2'JReturns the characteristc polynomial of Frobenius on the elliptic curve E.�@�@3'|����������ܜԜ��ȜlE4' __sig__3GDG\G5' ��М�����I6' XReturns the characteristc polynomial of Frobenius at p on the rational elliptic curve E.OOdO7'|��������������؜@R8' __sig__4T TtT9' �������V:'EThe theta series of the lattice with Gram matrix M to precision prec.0Z=' ��������[�'PhiLogetaS*True iff S is defined by a single equation��TS��������X]8\L]� V 8\T]����@ W IsHypersurface0 X��������P]Y=True iff Groebner basis calculations can be carried out for S�ZS��������@gp]8\d]�[  __sig__14\|\ 8\l]����x ] HasGroebnerBasis � � ^��������h]_ OTrue iff factorisation and irreducibility calculations can be carried out for SH%p%�%S��������Pg�]8\|]T)a  __sig__15�+�+b 8\�]�����-c HasFactorisation/�/�/d���������]e PTrue iff factorisation and irreducibility calculations can be carried out for S 5(5H5fS��������g�]8\�]4:g  __sig__16 ;;h 8\�]����\<i HasFactorization=�=�=j���������]k8True iff resultant calculations can be carried out for SDtD�DlS��������pg�]8\�]�Hm  __sig__17�K�Kn 8\�]����,Mo  HasResultantN Op���������]q2True iff GCD calculations can be carried out for S�STrS���������g�]8\�]4Xs  __sig__18�Y�Yt 8\�]����|[  EmptyScheme�;�;��������X\(Assign names N to the ambient space of S@[email protected][email protected] �S�����������fx\8\l\�D  __sig__1F�F�F  8\t\�����H 'The i-th name of the ambient space of S|L�L�L S����������f�\8\|\|P! __sig__2Q�Q�Q" 8\�\�����S#'The i-th name of the ambient space of SV(VXVS����������f�\8\�\�Z� Equation�����������T^�U��������x^p^^ � __sig__5 X p � ^l^����� � The degree of the hypersurface S(@�V��������n�^^t^t� __sig__6�� ^|^����,�The coordinate ring of S �  �S����������h�^^�^x!� __sig__7"�"�"� ^�^�����#�U����������^�^^ '� __sig__8())� ^�^�����+�The jacobian ideal of S�-�-�-�V����������^^�^�1� __sig__92�2�2� ^�^�����4�UD���������^�^^�9�  __sig__10|:�:� ^�^�����;�BThe matrix of partial derivatives of the defining polynomials of S�>�>�VD���������^^�^�B�  __sig__11�D�D� ^�^����,G�UD���������^�^^|L�  __sig__12�M�M� ^�^�����O�  HessianMatrix�P����������^� PThe matrix of second partial derivatives of the polynomial of the hypersurface SW�WX�VD���������^^�^�[�  __sig__13t]�]I2/usr/local/magma-2.5/package/AlgGeom/scheme/type.m������J%The scheme S as an affine plane curvep KUW��������4aa(at L __sig__0 L d M a0a�����N AffinePlaneCurveType�O��������,aP(The scheme S as a projective plane curvex�QVX��������Laa@a| R __sig__1 t � S aHa����|!TProjectivePlaneCurveType"# #U��������DaV9/usr/local/magma-2.5/package/AlgGeom/scheme/sch_closure.m�����W9/usr/local/magma-2.5/package/AlgGeom/scheme/sch_closure.mX������curveeXThe projective closure of S�/0YUV��������|ala\aa83Z __sig__04�4�4[ \aha����l6\ ProjectiveClosure9(9]��������da^(�A closure of the image of S under f whenever f is a standard chart up to a translation made by homogenising the equations individually[email protected]_[UV��������j�a\axa�D __sig__1F�F�Fa \a�a����Hb(�The intersection of the projective closure of S with the hyperplane at infinity in the projective closure of the ambient space of S�O�OcUV���������a\a�a(Sd __sig__2T�T�Te \a�a�����Vf SchemeAtInfinityX�X�Xg���������ah,The sequence of standard affine patches of S]^bWeights�� HessianDeterminant<K����������^�V��������� _^LO�  __sig__15�@� ^_�����W��������(__^�  __sig__16� ^_�����V� InflectionPoints@��7���������_� The points of inflexion of C�Y�X��������0_^_�  __sig__174\� ^,_�����W��������L_<_^��  __sig__18� � ^8_����� ���������4_� The points of inflexion of C � �X��������T_^H_�  __sig__19�� ^P_����t�WT��������p__^��  __sig__20� � � ^\_����h ���������X_�#One of the points of inflexion of C�#�XT��������x_^l_|'�  __sig__21<)T)� ^t_���� ,�0/usr/local/magma-2.5/package/AlgGeom/scheme/pt.m/ �� ���*0/usr/local/magma-2.5/package/AlgGeom/scheme/pt.m1p�� ���  coordinates 343�  equations�4�4�PT���������k�_�_�_�9� �_�_�����:Origin�;�T���������_�_�>� __sig__1@[email protected][email protected]� �_�_����B� HackobjCoercePt�D����������_�T���������_�_|L� __sig__2M�M�M� �_�_�����O�  HackobjInPt�P�P����������_� Print the point p at level lU�U�T�����������_�_�_�Y� __sig__3Z[([� �_�_����$u:/usr/local/magma-2.5/package/AlgGeom/scheme/hypersurface.m �����v:/usr/local/magma-2.5/package/AlgGeom/scheme/hypersurface.mx�����w7The hypersurface which is a common component of S and T� � xSS���������]�]�]�y __sig__0�(z �]�]����t� ^�^�����U���������_�^^� �  __sig__14� � � ^�^����D �P���������g�f�g� �f�g����(� IsProjectiveSpace@����������g�)True iff f is in the coordinate ring of A����P���������g�f�g��  __sig__16�� �f�g������ IsAmbientFunction����������g  coord_ringnction�=True iff f in is the function field (or coordinate ring) of A���P���������g�f�g�  __sig__17 � �f�g����x �IsAmbientRationalFunctionh*����������g�6/usr/local/magma-2.5/package/AlgGeom/ambient/amb_aff.mh��6/usr/local/magma-2.5/package/AlgGeom/ambient/amb_aff.mh��� ����������An n-dimensional affine space over k�C��Q��������(hhh hF __sig__0F(F hh�����N  AffineSpace�N��������h &An affine space with coordinate ring RXT �Q��������0hhh�W  __sig__1\P\  h,h����d�!The point of A with coordinates S\ \� __sig__0]�]�]�]^{ CommonComponent�|���������]}MTrue iff the hypersurfaces S and T do not have a common irreducible component� ~SS��������^�]�]�# __sig__1��� �]�]�����&� NoCommonComponent((x(����������]�8/usr/local/magma-2.5/package/AlgGeom/scheme/sch_common.m/�������,8/usr/local/magma-2.5/package/AlgGeom/scheme/sch_common.m20������U���������^ ^^|5� __sig__07 7d7� ^^�����9�The defining ideal of S(;@;X;�V���������a,^^ ^�>� __sig__1?@[email protected]� ^(^�����A�2The equations on some ambient space which define SXEpE�S���������<^^0^XK� __sig__2L�L�L� ^8^�����N�-The defining polynomial of the hypersurface S�P�S���������L^^@^�T� __sig__3V,V\V� ^H^����lX�-The defining polynomial of the hypersurface S,[� __sig__4\] ]� HackobjPrintPt�����������_� The sequence of coordinates of p � 8 �T���������_�_�_ � __sig__4 � � � �_�_����@�TS���������_�_t� __sig__5�� �_�_����,�True iff p is a point of the space A H TP���������_�_�! __sig__6"�"�" �_����"jTrue iff C is a sequence of coordinates of a point on S; if so also return the corresponding ambient point))ST�������_ �- __sig__7/�/�/ �_����L1 __sig__9ion2l2DTrue iff C is the coordinates of a point of S after some base change5x5S��������(�_ d:  __sig__8;<;T;  �_�����<  IsPointOverExtension=�= ��������   The origin of ABHB�BQT��������@�_4\G �_<�����I��������8 aThe sequence of coordinate points (0,..,1,..0) together with (1,..1) whether affine or projective<QTQP��������X�_L�U  __sig__10�VhW �_T�����XSimplexZ��������P  __sig__11^�^� __sig__4��� �bHc������'True iff p is not a singular point of S� � � �TS��������\c�bPc� � __sig__5 � � � �bXc�����,True iff the origin is a singular point of Sx�U��������lc�bc@� __sig__60H� �bhc����L �@The multiplicity of p as a point of S, where S is a hypersurface!T!|!�TS��������|c�bpc\� __sig__7%�%�%� �bxc���� '� __sig__9rity()�ETrue iff the tangent cone to S at p is reduced and S is singular at p�-�TS���������c�b�c�1� __sig__82�2�2� �b�c����l4� IsOrdinarySingularity�5����������c�TS���������c�c�b0<� �b�c����x=�  SingularLocus(?����������c�UU���������c�c�b�F�  __sig__10�G�G� �b�c����XK�LThe singular locus of S as an scheme embedded in the same ambient space as SOhO�VV����������c�b�cDR�  __sig__11TxT� �b�c����V�LThe singular locus of S as an scheme embedded in the same ambient space as SZdZ�SS���������c�b�c�^�  __sig__12H:The point whose coordinates are the negative of those of p��7TT��������p�_d�@  �_l���� Minus ��������h;Schemees "gThe nonzero coordinates of p as a sequence together with a sequence of the indices of these coordinates�@��7 T��������_|!  __sig__12��7" �_������@# NonZeroCoordinates4\���������%6The indices of the zero coordinates of p as a sequence�&T����������_�l '  __sig__13� � ( �_����� ) ZeroCoordinates *���������+6Return a random integral point of the ambient space A.(,PT����������_��-  __sig__14t . �_����� /  RandomPoint� � 0���������1�T�����������_X2  __sig__15�%�%3 �_����� '4 Polynomials defining pT)5T����������_� .6  __sig__16�/�/7 �_������18�TS�����������_�49  __sig__17 6h6: �_�����9<���������=8The reduced scheme supported on argument point or points<=,=>TS��������tda�_�A?  __sig__18�B�B@ �_�����<EA�T���������a a�_TJB  __sig__19dL|LC �_a�����MD>The ideal defining the reduced scheme structure supported on pQET��������� a�_a0UF  __sig__20lV�VG �_a�����XH2/usr/local/magma-2.5/package/AlgGeom/scheme/type.m��P�M HackobjInSchProj� N��������(eO(Print the projective scheme S at level l � � PV����������He�d<e Q __sig__5PhR �dDe�����S HackobjPrintSchProjHtT��������@eU The affine cone of S�VVU���������ie�dTeh W __sig__6 h X �d\e�����!Y  AffineCone##Z��������Xe[5/usr/local/magma-2.5/package/AlgGeom/scheme/support.m���\ 5/usr/local/magma-2.5/package/AlgGeom/scheme/support.m���]T���������e|epe�/^ __sig__0141L1_ pexe�����2U���������e�epeh6a __sig__18�8�8b pe�e����d:cGThe sequence of ambient points which support S if it is a finite scheme�<�<�<dV���������n�epe�e�@e __sig__2BHB�Bf pe�e�����Dg"jA sequence of maps from affine spaces whose images partition P together with the additional point returned�KLhRT�������epe�e�Oi __sig__3P�P�Pj pe�e����@Rw __sig__0nsiontTkU���������e�epehXl __sig__4YZ0Zm pe�e�����[n HasPointsOverExtension�]o���������ep@False iff the support of S is defined over the ground field of Sd�d�d* HackobjCoerceSchAffH+���������d,ITrue iff the point x lies on S or the sequence of coordinates x lies in S� � -U���������dTd�dh. __sig__4�/ Td�d����(0 HackobjInSchAff�1���������d2Print the affine scheme S at level l�3U�����������dTd�d� 4 __sig__5!�!�!5 Td�d����H#6 HackobjPrintSchAff��7���������d86/usr/local/magma-2.5/package/AlgGeom/scheme/sch_proj.m"�96/usr/local/magma-2.5/package/AlgGeom/scheme/sch_proj.m"�; __sig__0eterminant��� ���������������������������:�RV���������d�d�d�8< �d�d����d:=RV��������e�d�d�<? �d�d�����>@FThe subscheme of the projective space P defined by the given functions�BA�RV�������� e�de\GB __sig__2H�HIC �de����LLK __sig__4rojM|MDV��������e�dQE __sig__3QR@RF �de�����TG HackobjCoerceSchProjVXVH��������eIITrue iff the point x lies on S or the sequence of coordinates x lies in SD]\]JV��������0e�depaL �d,e�����c  RemoveHypersurface�@��7 ��������d*The scheme S with any Z components removed�@SSS��������Dd�c8d __sig__4�@��7 �c@d����  Difference��7��������<d5/usr/local/magma-2.5/package/AlgGeom/scheme/sch_aff.m�%� 5/usr/local/magma-2.5/package/AlgGeom/scheme/sch_aff.m�%� is_hypersurface�@ singular_locus��� is_empty� __sig__2X�� �b�b���� �U���������b�b�b � __sig__3 �  � �b�b����4�GTrue iff S is defined by the function 1 or by the trivial maximal ideal,Lx�V���������b�b�b@� __sig__40H� �b�b����L � True iff S is contained in T � �SS���������b�b�bL#� __sig__5$$$\$� �b�b����(&�9/usr/local/magma-2.5/package/AlgGeom/scheme/singularity.m@(�(.��9/usr/local/magma-2.5/package/AlgGeom/scheme/singularity.m�(�(.��T�������� cc�b�1� __sig__02�2�2� �b�b����l4�  IsNonSingular|5����������b�U�������� cc�b<� __sig__1<�<=� �bc�����>�-True iff S is nonsingular and equidimensional�A�V��������Tc$c�bc�F�
__sig__2G�G�G��b c����XK�DTrue iff S has at least one singular point or is not equidimensionalNO�S��������Dc4c�b(cR�
__sig__3S�ST��b0c�����U�

IsSingular�W�W���������,c� True iff p is a singularity of S\h\�\�TS��������Lc�b@c�iV���������j�a\a�a�j
__sig__3	�	�	k\a�a����dl

AffinePatchestm���������an,The sequence of standard affine patches of S�oV���������j�a\a�axp
__sig__4�@q\a�a������S��������4b$b(b�� $b0b�����	�True iff S is reduced(�S����������Db$b8b� � __sig__1��� $b@b������7The support of the reducible part of the hypersurface S���S��������Tb$bHb�� __sig__2  x � $bPb������
ReducibleSchemex!���������Lb�True iff S is reducible�$�$�$�S��������lb$bb�(�
__sig__3*�*�*�$bhb�����,�  IsReducible�. /���������db� True iff the dimension of S is 02�2�2�S���������b$bxb�6�
__sig__48�8$9� $b�b�����:�8/usr/local/magma-2.5/package/AlgGeom/scheme/comparison.m<�5��9��8/usr/local/magma-2.5/package/AlgGeom/scheme/comparison.m?6��9��
__sig__1pace�A�8True iff the two arguments lie in the same ambient spaceETElE�SS���������b�b�bTK�
__sig__0L�L�L��b�b�����N�IdenticalAmbientSpaceLP����������b�UTrue iff S and T are defined by the same ideal of functions on the same ambient space�V�SS���������g�b�b�b�Z��b�b����]�T���������b�b�b8ar
AffineCharts�s���������at#The i-th standard affine patch of S�"�"uVU��������b�a\a�a$&v __sig__5&�& 'w \a�a����T)x  AffinePatch�+�+y���������az#The i-th standard affine patch of S�0�0{VU��������b�a\a�a84| __sig__65(5H5} \a�a����7~  AffineChart�9�9���������a� ]A standard affine patch of S containing p, the point p in that chart and the chart map itself$?�TVUT[�����jb\abxC�
__sig__7DE<E�\ab����\G�]A standard affine patch of S containing p, the point p in that chart and the chart map itself,M�TVUT[�����jb\ab�P�
__sig__8Q�Q�Q�\ab����tT�8/usr/local/magma-2.5/package/AlgGeom/scheme/firstprops.mWP?��@��
8/usr/local/magma-2.5/package/AlgGeom/scheme/firstprops.mZ�?��@��=True iff the hypersurface S is irreducible over its base ring_�
__sig__0t���b�c����$�P��������Hg�f<g� �fDg����(�NTrue iff factorisation and irreducibility calculations can be carried out in At�P��������Xg�fLg�� �fTg������NTrue iff factorisation and irreducibility calculations can be carried out in A��P��������hg�f\g�  __sig__10(� �fdg����@�7True iff resultant calculations can be carried out in Ahl�P��������xg�flgH �  __sig__11l � �ftg����T*�1True iff GCD calculations can be carried out in A�*�P���������g�f|g8�  __sig__128� �f�g����,8�,True iff A and B are the same ambient objectP8�PP���������k�g�f�g�C�  __sig__13F� �f�g����,F� True iff A is an affine space�N�P���������g�f�g@T�  __sig__14TT� �f�g����W�  IsAffineSpace,W����������g� True iff A is a projective spacedhd�  __sig__15�Q �QU���������d|dTd������������������ ��������������������������  __sig__0��  Tdxd�����b� SingularScheme�����������c�;/usr/local/magma-2.5/package/AlgGeom/scheme/constructions.m�J�XS�� ;/usr/local/magma-2.5/package/AlgGeom/scheme/constructions.m�J�XS��>The scheme defined by all of the defining equations of S and T��SSS���������c�c�c�� __sig__0\|� �c�c����x �>The scheme defined by all of the defining equations of S and Tx!�SSS��������d�c�cX$
__sig__1%�%�%�cd����'.The scheme S with its reduced scheme structure�*SS��������d�cd|/
__sig__20�0�0�cd����<2

ReducedSchemeh3��������d,�The scheme S after removing any T components of it assuming that S and T are both hypersurfaces; the number of factors removed is also returned�;<,<	SSS������,d�c d�?

__sig__3@A$A  �c(d����xC hessian_determinant��  projective_closure_map� hessian_matrixTK QU���������d�dTddO" Td�d�����P#BThe subscheme of the affine space A defined by the given functions TtT$�QU���������d�dTd�dhX&Td�d����Z'U���������dTd�^(
__sig__3D\)Td�d�����a�TUU��������$f f�e�� __sig__4 T l � �ef����� �IThe tangent cone to S at p embedded as a scheme in the ambient space of Sx � �TVV��������,f�e f� __sig__5�$��e(f����t�JThe tangent bundle of S together with the projection map of ambient spaces���UU1������<f�e0f� �
__sig__6!""��e8f����x#�

TangentBundle�$���������4f�1The subscheme of S whose tangent spaces contain p�)*�TUU1������Tf�eHfL/� __sig__70�0�0� �ePf���� 2�  Tangencies 343���������Lf�*True iff p lies on some tangent space of S�8$9�TU��������lf�ef�;�
__sig__8<�<�<��ehf�����>�

IsTangency[email protected][email protected]���������df�6/usr/local/magma-2.5/package/AlgGeom/scheme/pt_tests.m�\��6/usr/local/magma-2.5/package/AlgGeom/scheme/pt_tests.m�\��'True iff the points of S line on a line|M�M�M��T���������f|f�fTQ�
__sig__0R�R�R�|f�f�����T�
AreCollinearVXV����������f�2/usr/local/magma-2.5/package/AlgGeom/ambient/amb.m�[��\��"2/usr/local/magma-2.5/package/AlgGeom/ambient/amb.m�[��\��
nr_of_coordsdHd�IsCurveqV���������epe�e�r
__sig__5	�	�	spe�e����(t5/usr/local/magma-2.5/package/AlgGeom/scheme/tangent.m�hf�u5/usr/local/magma-2.5/package/AlgGeom/scheme/tangent.m�hf�vTTT���������e�e�etx�e�e����\�&t����������d����&d�������(�&"Returns the eigenvalues of M on V.H�&t������������d�����&
__sig__4��&d��������'
__sig__0ent��&SSpace orthogonal to u with respect to the inner product given by the Gram matrix M.��&ut��������ԛěd���,�&
__sig__5@�&d�������X�&OrthogonalComplementp�&�����������& _Space orthogonal to elements of S with respect to the inner product given by the Gram matrix M.T*\*�&�ut���������ܛd�Л�*�&
__sig__61�7�&d�؛����8�&SSpace orthogonal to U with respect to the inner product given by the Gram matrix M.@8�&tt����������d���C�&
__sig__7C�C�&d������F�&JSubspace of V orthogonal to u with respect to the Euclidean inner product.�N�&utt�����������d��HT�&
__sig__8T\T�&d������� W'VSubspace of V orthogonal to elements of S with respect to the Euclidean inner product.�_'�utt����������d���ffunction_field�����������������calc_factorisationZ4Z�
calc_resultant�[���������
calc_gcd_D_h_�Assign names N to A�<a��P�����������f�f�f�e�&-/home/was/magma/Kohel/hecke_decompositions.mg�8r��&-/home/was/magma/Kohel/hecke_decompositions.mg�8r��&KReturns the sequence of vector subspaces stabilized by the Hecke operators.Xp�&��������,�� ��&
__sig__0��&�(�����$�&HeckeModuleDecomposition���&��������$��&*�Returns the sequence of one dimensional vector subspaces stabilized by the Hecke operators, priority given to the divisors of the level N.�!�!�&��������D��8��$�& __sig__1&<&T&�&  �@������'�& [Returns the sequence of one dimensional vector subspaces stabilized by the Hecke operators.�-�-�& ��������@�T� �H��1�& __sig__22�2�2�&  �P�����h4"% ���������� ���$9�&HeckeEllipticData|:�:�&�����������&'/home/was/magma/Kohel/linear_algebra.mg8r��&'/home/was/magma/Kohel/linear_algebra.mg8r��&&An LLL reduced basis of the span of B.xC�&�9��������|�t�d�h��G�&
__sig__0I0JTJ�&d�p������L�&6An LLL reduced basis of the sublattice generated by B.�O�&������������d�x�$S�& __sig__1T�T�T�& d��������V�& __sig__3ynomialX�&0Returns the characteristic polynomial of T on V.[L[|[�&t ��������̜��d���,�& __sig__2aPapa�& d��������c�&)Returns the minimal polynomial of T on V.�f4g�  calc_groebner�� __sig__0 � � � �f�f����h � The i-th name of A�!�!�P����������f�f�f�$�
__sig__1&<&T&��f�f�����'�The i-th name of A�)*�P����������f�f�fL/�
__sig__20�0�0��f�f����2�0The base ring of A iff A is defined over a field�P����������Hu�f�f�f9�
__sig__5f�f�����The base ring of A<,<�P����������g�fg�?�
__sig__4@A$A� �f g����xC� �fg����lE� The dimension of AtG�G�P���������} g�fg�L� __sig__6M�M�N� �f g����LP�)The number of coordinates of a point of AR@R�P��������0g�f$g�V�
__sig__7XLXhX��f,g����Z�NumberOfCoordinates\\���������(g�<True iff Groebner basis calculations can be carried out in AaXb�
__sig__8d0dHd�
__sig__9e�e@fy
TangentSpaceE�Fz���������e{TUU���������e�e�e�M|
__sig__1O�O�O}�e�e�����P~JThe tangent space to S at p embedded as a scheme in the ambient space of S�T�TTVV��������f�e�e�X�
__sig__2ZtZ�Z��ef�����\�TTT��������ff�e��
__sig__3a�aXb��ef�����d�deHsign_decomposition_plus ˸�K
is_canonicalIsign_decomposition_minus(J
is_effective<Mnormal_form
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is_principal�^X]^���������m�mmtm�_
__sig__3�m|m�����a.The prime divisor consisting of the place at p�bT]^���������m�mm�m$c __sig__48d m�m����Pe(�The principal divisor (line through p,q) - (line at infinity) restricted to the curve together with the defining rational functionH fTT]^��������mm�m*g __sig__5*t*h m�m�����*v __sig__8rveElt1i^���������mm 8j __sig__6848k m�m����L8lHackobjCoerceDivCurveEltC�Cm���������mn Print d at level l Fo^�����������mm�m�Np __sig__7N�Nq m�m����@Tr HackobjPrintDivCurveEltXTs���������mt The divisor group containing d�Wu^]���������mm�mpdw m�m�����fx The prime factorisation of dkz __sig__9ll~  __sig__108X Print D at level l Z4Z]����������Xl�kLl^ __sig__3_�_0 �kTl�����a=&���������>&$uReturns the leading p coefficients of the nth division polynomial defining the kernel of the multiplication by n map.?&|o���������Ȗ��8A&Ȗ�����dB&PuiseuxDivisionPsi�C&����������D&$rReturns the function field element defining the numerator of the multiplication by n map on the x-coordinate of E.�E&|o�������� �Ȗ�F&  __sig__27 G& Ȗ�����4H&  DivisionPhiHI&���������J&$rReturns the function field element defining the numerator of the multiplication by n map on the y-coordinate of E.hK&|o��������4�Ȗ(�p*L&
	__sig__28�*M&Ȗ0�����l1N&

DivisionOmega�7O&��������,�P&NReturns the function field elements defining the multiplication by n map on E.P8Q&|o��������L�Ȗ@��CR&
	__sig__29FS&ȖH�����,FT&DivisionPolynomials�NU&��������D�h&
	__sig__33mialnTV&@Returns the classical modular equation relating j(q) and j(q^N).WWW&���������t�d�ȖX�P\X&
	__sig__30\dY&Ȗ������fZ&ClassicalModularEquationhh^&
	__sig__31�IH'True iff x is a point or subscheme of A�H IR��������,i�h i�
K�h(i������&&/home/was/magma/Kohel/modular_forms.mg���&'The cusp form of a minimal model for E.Ȗ�������&����������&AReturns the j-invariant as an element of the function field of X.sical Eise�&�������������$����& $��������&8Returns the canonical involution of the modular curve X.��������&EisensteinFunction$%L%�&�����������&���������К$�̚���&
__sig__8�&$�Ԛ����Ȗ�&6�Returns an integer representing the "genus type" of X, defined to be 0 if the genus of X is zero, and 1 if X is of genus one or hyperelliptic over the quotient by the canonical involution.�&����������$�ܚ8�&
__sig__9�������&CanonicalInvolution(>�>�&��������Ț�&*�Returns an element of the function field which disappears on the locus of points for which this model of the modular curve is singular.iptic over t�&�������������$���&  __sig__10��Ȗ�& $��������P�&	__sig__11��R�&
	GenusType�ST�&��������ؚ�&%Returns the modular curve of level N.ld whi�&����������$� �]�& $�������&|����������������8��$' __sig__0Pe % ������p��(���ʸ The affine plane over k�5$6l6��Q��������@hh4h�:
__sig__2;�;�;h<h����=!The point of A with coordinates S\?|?QT������PhhDhD
__sig__3EXEpEhLh�����GHackobjCoerceAmbAff4JXJ��������Hh'True iff x is a point or subscheme of A�OP PQ��������hhh\h�S
__sig__4TU4Uhdh����lW
HackobjInAmbAff�X��������hPrint A at level lH]]Q�����������hhthta
__sig__5bc<c h|h����Pe[&��������\�\&GReturns the classical modular equation over K relating j(q) and j(q^N).�	�	8
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_&Ȗx�����0&+Returns the Hilbert class polynomial for D.�a&����������Ȗ���b&
	__sig__32l�c&Ȗ�������d&HilbertClassPolynomial�e&����������f&$uReturns the sequence of reduced forms for the order of index m in the quadratic field of fundamental discriminant DK.H%g&����������Ȗ��$)i&Ȗ�������+j&4Returns the inverse of f as a binary quadratic form./L/k&����������Ȗ��l2l&
	__sig__34�3�3m&Ȗ������H5n&
FormInverse�67o&����������p&,�Returns the reduced binary quadratic forms in the kernel of the map of class groups from forms of discriminant m^2*DK to form of discriminant DK.X?x?q&��������̙Ȗ��\Dr&
	__sig__35TElEs&Ȗș�����Gt&
KernelForms�I�Iu&��������ęv&>Returns the composite of the binary quadratic forms q1 and q2.LPw&���������ȖؙTx&
	__sig__36XU�Uy&Ȗ������Wz&
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FormComposite�X{&��������ܙ|&8Returns the reduction of the quadratic form A = <a,b,c>._@_d_}&����������Ȗ��c~&
	__sig__374eLe&Ȗ�������g�&
	__sig__38pi�i�QR��������j�ij��
__sig__3ity	�	�HThe hyperplane at infinity as a subscheme of the projective closure of A���QV��������j�ij��
__sig__2(���ij������HyperplaneAtInfinity���������j�>The line at infinity as a curve in the projective closure of A�QX��������4j�i(j�"��i0j����$� LineAtInfinity�%���������,j�*The sequence of standard affine patch maps@+�+�R��������Lj�i@j�/� __sig__40�0 1� �iHj����l2� AffinePatchMaps�3���������Dj�*The sequence of standard affine patch maps�9�9�R��������dj�iXj\<� __sig__5=\=t=� �ij����x?����������j�@The sequence of affine spaces that comprise the standard patchesF�F�F�R���������j�i�jL� __sig__8LM,M� �i�j����dO�@The sequence of affine spaces that comprise the standard patchesQ�Q�Q�R���������j�i�jXV� __sig__9X X4X� �i�j����0Z�#The i-th standard affine patch of Pd\�\�RQ���������j�j�i�j��  __sig__10�aXb� �i�j�����d�#The i-th standard affine patch of P�g�g�  __sig__11pi�i]��������di^/The sequence of weights on the coordinates of P��7_R���������i�hxi��7 __sig__9\a �h�i����c��������|id'False iff the grading on P is [1,1,...]�@��7eR���������i�h�if  __sig__10��7g �h�i�����@h  IsWeighted7i���������ijThe function field of P! HackobjPrintAmbAff� "��������xh#The coordinate ring of A � 8$Q���������Ti�hh�h
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__sig__7t*h�h����+7/usr/local/magma-2.5/package/AlgGeom/ambient/amb_proj.m ��,7/usr/local/magma-2.5/package/AlgGeom/ambient/amb_proj.m ��-
gradingng�#$. affine_patch_mapsx�� ��/ affine_cone_map�&�& '01An n-dimensional projective space over the ring k@+�+1��R���������h�h�h�h�/2 __sig__00�0 13 �h�h����l2�&;Returns the discriminant of the quadratic form q = <a,b,c>.���&�������� �Ȗ�� �& Ȗ�����@ �& bReturns the class number of the order of discriminant D in a quadratic extension of the rationals.��&�������� �Ȗ���&  __sig__39<\�& Ȗ����� �&1Returns the points of the projective line over R.0 h �&M��������,�Ȗ �X"�&  __sig__40#x#�& Ȗ(������$�&
ProjectiveLineT&�&��������$��&8Returns the sequence of maximal prime powers dividing n.,�,�,�&��������D�Ȗ8� 1�&  __sig__41$2<2�&Ȗ@������3�&PrimePowerDivisors(5H5�&��������<��&Returns the set of units in R.�:�&M����������\�ȖP�,=�&
	__sig__42�>�>�&ȖX������@�&��������T��&
__sig__6ldE<E�&)Returns the level of the modular curve X.curve X.�&���������p�$�l�Ȗ�& $�t������O�&$��������P�& ModularFunctionFieldR@R�&�����������&(�Returns the function field element giving the quotient of the classical Eisenstein series of weight k by a power of a fixed form on X.�& __sig__7˸Ȗ�&  ModularLevel_,�&��������h��&&/home/was/magma/Kohel/modular_forms.mg���*' __sig__1�g�g�&  ModularCurveiHiL HackobjInAmbProjHM��������$iNPrint P at level l�	8
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__sig__7��W�hXi�����!X'The grading on the coordinate ring of Px#�#�#YR��������li�hi�&Z
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	__sig__11�01m�h�i����l2n6The affine cone of P together with the projection to P5oRQ[�������i�h�i�9p
	__sig__12�:�:q�h�i����,<r&The map from the affine cone of P to P�=sR[���������i�h�iBt
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AffineConeMap�Gw���������ix:/usr/local/magma-2.5/package/AlgGeom/ambient/amb_closure.m������y:/usr/local/magma-2.5/package/AlgGeom/ambient/amb_closure.m0������
__sig__1MapS�Sz2The inclusion map from A to its projective closurelV�V{Q[���������i�i�i�Z|
__sig__0\d\�\}�i�i�����^~ProjectiveClosureMap����������i�The projective closure of Ade�e��ij����Th4
ProjectiveSpace�45���������h65A projective space with homogeneous coordinate ring R�:7�R���������h�h�h�h,=8
__sig__1>�>�>9�h�h�����@:1A weighted projective space over k with weights WtD�D;��R���������h�h�h�H<
__sig__2K�K�K=�h�h����,M>The projective plane over k�O�O?��R��������i�h�h�R@
__sig__3T�T�TA�hi����XVJ
__sig__5rojX4XB!The point of P with coordinates StZ�ZCRT������i�hi_D
__sig__4t�E�hi����XbFHackobjCoerceAmbProjd�dG��������i�RQ���������j�j�i�j���i�j�����	�]A standard affine patch of P containing p, the point p in that chart and the chart map itself
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	__sig__12�(��i�j����t� A standard affine patch of P containing p, the image of p in that chart and the chart map itself���TRQT[�����j�i�j� �
	__sig__13""��i�j����x#�5/usr/local/magma-2.5/package/AlgGeom/divisors/place.m�����5/usr/local/magma-2.5/package/AlgGeom/divisors/place.m�����
	is_simple@+�+�projective_curve,-\-�point/�
���puiseux_expansion�1�1�TW_��������8k(kk5�
__sig__06�6�6�k$k�����9�  SimplePlace|:�:��������� k� Create the simple place p on C�=�TX_��������@kk4k�A� __sig__1CDD\D� k<k�����F� __sig__4rve�G�_��������Lkk�L� __sig__2M�M�N� kHk����LP� HackobjCoercePlcCurve�Q���������Dk� Print pl at level l�UV�_����������dkkXkZ� __sig__3[�[�[� kk�����]� HackobjPrintPlcCurve_,���������\k�8True iff pl is a place at a nonsingular point of a curvefXfpf� kxk����i� __sig__5j8jPjhj�j�%$Returns the identity isogeny E -> E.��%|����������Ȗ�<
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	__sig__16�2�2	&Ȗ �����l4
&EllipticEndomorphism5�5&���������&:Returns the pushforward of the kernel polynomial psi by f.�;�;
&�oo��������<�Ȗ0�(?&Ȗ8������@&
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	__sig__18�O�O&ȖP�����(Q&
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	__sig__22ies�a&&{Returns the Tate model for the Weierstrass p-series, in terms of q = exp(2*pi*i*tau) and w = exp(2*pi*i*z), to precision N.�hi&����������������|�Ȗp�mA
BestHyperplane�B���������lC6/usr/local/magma-2.5/package/AlgGeom/divisors/divelt.m���DH6/usr/local/magma-2.5/package/AlgGeom/divisors/divelt.m���G
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	__sig__20�&Ȗx����� &TateWeierstrassSeries !&��������t�"&"jReturns the Tate model for the Weierstrass p-series, in terms of q = exp(2*pi*i*tau) and w to precision N.�#&����������������Ȗ���$&  __sig__21�%& Ȗ�������&& ]Returns the formal group power series x(z) of E, followed by the power series y(z) = -x(z)/z.$'&|������������Ȗ��X)&Ȗ������p*&
FormalGroup�+&����������,&[Returns the function field element defining the kernel of the multiplication by n map on E.l*-&|o����������Ȗ��p1.&
	__sig__238/&Ȗ������80&
DivisionKernel,81&����������2&$vReturns the nth division polynomial, a function field element defining the kernel of the multiplication by n map on E.F3&|o��������ԘȖȘ�N4&  __sig__24�N5& ȖИ�����N6&  DivisionPsiHT7&��������̘@&  __sig__26si(W8&$uReturns the leading p coefficients of the nth division polynomial defining the kernel of the multiplication by n map.ld9&|o���������Ȗ�k:&
	__sig__25l;&Ȗ�����m<&TruncatedDivisionPsi������
AffineChartMapsPL���������\j�"The i-th standard affine patch map�P�P�R[��������|j�ipj�T�
__sig__6U�UV��ixj����8X�
AffinePatchMap�Y���������tj�"The i-th standard affine patch map_ _�R���������j�i�j<c�
__sig__7d�d e��i�j�����g�
AffineChartMapLiy^�����������mm�m�{m�m�����	|!The curve on which d is a divisorx�}^S��������4p�mm�m0m�m������True iff d is the zero divisort�^��������nmn<�
	__sig__11,D�mn����H�True iff a is equal to b���^^��������tpnmnH#�
	__sig__12 $X$�mn����$&�  Degree of d�& '�^�������� q,nm nX,�  __sig__13�-�-� m(n�����/�.True iff all coefficients of d are nonnegative<2�^����������<nm0nx5�  __sig__1477� m8n�����9�  IsEffective�:�:���������4n�3Effective minimal divisors a,b satisfying d = a - bX?x?�^^^������TnmHn\D�  __sig__15TElE� mPn�����G� SignDecomposition0JTJ���������Ln�The sum of divisors a + b�O�O�^^^�������� qlnmn�R�  __sig__16�T�T� mhn����XV� The inverse of dXLXhX�^^���������n|nmpn�\�  __sig__17�]^� mxn����\� The difference of divisors a - bbpb�b�^^^��������,q�nm�n�g�  __sig__18pi�i� m�n����hk�  __sig__19�lm$representative_divisor�%&The linear equivalence class of d in J&^a���������p�p�pH'
__sig__0t(�p�p�����6
__sig__3rveElt)a���������p�p�*
__sig__1�+�p�p�����>(The curve on which d is a jacobian point?aS��������q�p�pDA�pq����\B
Degree of jpCa���������qq�pq\D
__sig__6pE�pq����\*Fj + kl*Gaaa��������$q�pqp1H __sig__778I �p q����8J -j(8K aa��������<q4q�p(q\8L __sig__8C�CM �p0q�����CN j - kFO aaa��������Dq�p8q�NP __sig__9N�NQ �p@q�����NR n*jHTS aa��������Tq�pHq(WT  __sig__10�WU �pPq�����_X JThe uniformizing element (an element with valuation 1) of the local ring L�fY ����������lrlq\qqlZ __sig__0m m[ \qhq����4m� mlistO normal_form_rational_functionN normal_form_divisor�@S __sig__0  P  linear_systemF  �    QHThe divisor in the divisor group D given by the factorisation sequence S�@R�]^��������XmPmmDmT mLm����U,The divisor in D of the ambient polynomial f�@V�]^��������hmmmTmW __sig__1�@��7X m\m����Y5The divisor in D of the ambient rational polynomial fZ�]^��������xmpmmdm[ __sig__2h\ mlm����k] The divisor in D of the line L� ProjectiveCurve�����������k�The point underlying pl� � � �_T���������kk�k� � __sig__7 � � � k�k�����The point underlying pl(��_T��������4o�kk�k\� __sig__8,� k�k����� � UnderlyingPointh ����������k�True iff pl is a place of C�"�"�_S���������kk�k$&�
__sig__9&�&'�k�k����T)����������k�5True iff p and q are the same place on the same curve�/�__���������l�kk�k43�
	__sig__10�4�4�k�k����h6�3/usr/local/magma-2.5/package/AlgGeom/divisors/div.m8�P��3/usr/local/magma-2.5/package/AlgGeom/divisors/div.m��P�
affine_model=,=
affine_curve>�>Egroupzeroctive_c
__sig__0has_normal_formsDE<Ebest_hyperplane,GDG\GW]��������(l l�k�N�kl����LP	The divisor group of C�Q
X]��������0l�k$l�U  __sig__1W�W�W  �k,l����tY __sig__4rve�Z 3Consider the nonsingular point x as a prime divisor�]^]^������@l�k4lXb __sig__2d0dHd �k<l����pf HackobjCoerceDivCurvei��������8l�_��������|kkpk� The curve of which pl is a place�_S���������l�kk�k� k�k�����+The projective model of which pl is a place�_X���������l�kk�k� __sig__6� k�k���� HackobjPrintDivCurve ��������Pl,True iff D was created using an affine curve � ]��������pl�kdl  �kll�����  FromAffineCurve� ��������hl &The curve containing the elements of D ]S���������m�l�k|l ! __sig__5 � � " �k�l����� # The affine model underlying D�"$]W��������Dp�l�k�l�%%
__sig__6&�&�&&�k�l����$)'  AffineCurve@+�+(���������l)1The projective curve containing the elements of D�0 1*]X��������Tp�l�k�lh4+ __sig__755x5, �k�l����t8- The zero divisor9�94:.]^��������dp�l�k�l�</ __sig__8=�=�=0 �k�l����,@1  ZeroDivisorpA�A2���������l3True iff D is equal to EFG,G4]]��������n�l�k�l|L5 __sig__9M�M�M6 �k�l�����O7ATrue iff D is associated to a cubic curve with a flex at infinity�R�R8]^�������l�k�lhW9  __sig__10�X�X: �k�l�����Z; HasNormalForms \<���������l=6A good choice of divisor corresponding to a hyperplane�a>]^��������m�k�l4g?  __sig__11�hi@ �k�l�����j�Omitces4# ssbasis�5# weights  !3# 25]^_ 7#eisen�"�8#8�����������8sL9#  __sig__311UV ����������t8sTt� __sig__0� 4 8s\tXx � ManinSymbolApply� ��������Xt dpApply an element g=[a,b, c,d] of SL(2,Z) to the given modular symbol. The definition of the action is g.(X^i*Y^j{u,v}) := (dX-bY)^i*(-cX+aY)^j {g(u),g(v)}. A modular symbol is represented as a pair <P,x>, where P is a polynomial in X and Y (an element of the ring MmlistR, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.*l* t���������txt8sltp1 __sig__178 4 8sttX� 8 ModularSymbolApply08 ��������pt! f}Apply an element g=[a,b, c,d] of SL(2,Z) to the given modular symbol. The definition of the action is g.(X^i*Y^j{u,v}) := (dX-bY)^i*(-cX+aY)^j {g(u),g(v)}. A modular symbol is represented as a sequence of pairs <P,x>, where P is a polynomial in X and Y (an element of the ring MmlistR, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.\T" t���������t8s�t�W# __sig__2\�_$48s�tX�dd%[Returns the space of modular symbols of weight 2 for Gamma_0(N), over the rational numbers.k&t���������t�t8s�t$m' __sig__3m8m( 8s�t����Ds ���������� rr\q�q�� __sig__8 X p � \qr����� � 3The unramified local ring with prime p and degree f�  � ��������� rr\qr�� __sig__9�,� \qr����x� @The unramified local ring with prime p, degree f and precision nx� ���������,r$r\qr��
	__sig__10!<!�\q r�����"�SThe totally ramified local ring with prime p defined by the Eisenstein polynomial g�%(&����������<r4r\q(r*�
	__sig__11<,\,�\q0r���� .� cThe totally ramified local ring with prime p and precision n defined by the Eisenstein polynomial g�2�2����������LrDr\q8rl6�
	__sig__12�8�8�\q@r����h:�WThe local ring with prime p and inertia degree f defined by the Eisenstein polynomial g0==x=����������\rTr\qHr�A�
	__sig__13DC|C�\qPr����pE� dThe local ring with prime p, inertia degree f and precision n defined by the Eisenstein polynomial gLPL����������dr\qXr P�
	__sig__14Q(Q�\qr�����R�JThe uniformizing element (an element with valuation 1) of the local ring LWlW�����������tr\qhr,[�
	__sig__15] ]�\qpr���� _�*�The known part of the local element x. Note that this function will be removed in the next release; Expand() should be used in its place.�eDf������������r\qxr�j�
	__sig__16lxl�\q�r����To�eps3�,HackobjCoerceJacCurveElt@V-���������p.Print j at level l�W/a�����������p�p�p0
__sig__2d[1�p�p����2HackobjPrintJacCurveElt��73���������p4The jacobian containing j�75a���������p�p�p��77�p�p����@
__sig__5isor�78A divisor in the class of j��79a^���������p�p�p�@:
__sig__4��7;�p�p������7<RepresentativeDivisor=���������p�JThe underlying point in the projective plane of the single place divisor d���^T��������<om0o�m8o������1The sequence of points underlying the places of d���^��������Lom@o$�  __sig__29�� mHo����<� UnderlyingPoints\|���������Do�EThe sequence of points forming the support of the effective divisor d"�^��������domXoH%�  __sig__30l&�&� mo����t(�  SupportPoints*���������\o�$vA divisor linearly equivalent to d but in a normal form (only in the case that the curve is a nonsingular plane cubic)l2�^^��������|ompo�5�
	__sig__31�7t8�mxo����4:�#Degree 1 divisor corresponding to d�;�;�^^���������om�o$?�  __sig__32�@�@� m�o�����B� NormalFormDivisor�D�D����������o�  __sig__34Function� [A rational function on projective plane whose divisor plus the normal form of d is d itself$OdO�^����������om�o@R�
	__sig__33 TtT�m�o����V�NormalFormRationalFunctionLXhX����������o�ZTrue iff d is a principal divisor in which case also return a rational function defining d�_,�^���������om�oHd�m�o����pf�
� eTrue iff d and e are linearly equivalent in which case also return a rational function defining d - eXn�The divisor n*d H�^^��������Lq�nm�n�	�m�n�����:The sum of the norms of the finite place coefficients of dx
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	__sig__20�$� m�n����t�EThe sequence of places forming the support of the effective divisor d��^���������nm�n� �  __sig__21�!�!� m�n����H#�=The underlying place of the divisor d based on a single point$&�^T���������nm�n*�
	__sig__228,X,�m�n����.�
UnderlyingPlace�/����������n�FThe sequence of places appearing in the factorisation of the divisor d5�^���������nm�n�9�
	__sig__23�:�:�m�n����,<�UnderlyingPlaces=\=t=����������n�_^^��������o�nm�D�
	__sig__24�F�F�m�n����H�"The component of d associated to p4LLL�T^^��������om�nP�
	__sig__25Q$Q� mo�����R�_^^��������$oomhW�
	__sig__26�X�X�mo�����Z�%��������,��%|��������L�Ȗ@�<
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__sig__5Ld�%ȖH�������%6Returns true if P satisfies the equation of the curve.��%~|��������l�\�ȖP�(�%
__sig__6�%ȖX�����@�%
	IsOnCurve0H�%��������T��%6Returns true if P satisfies the equation of the curve.|!�%|��������t�Ȗh�\$�% __sig__7%�%�%�% Ȗp����� '�%@Returns the value of the defining equation of E, evaluated at P.+,$,�%~|������������Ȗx�p0�%
__sig__81�1�1�%Ȗ�������2�%EquationEvaluation�4�4�%��������|��%@Returns the value of the defining equation of E, evaluated at P.:�:�:�%|����������Ȗ���=�%
__sig__9?\?|?�%Ȗ������(A�%0Returns the value of f evaluated at the point P.D�D�D�%~o��������������Ȗ���I�%
	__sig__108LPL�%Ȗ�������M�%FunctionEvaluation�O�O�%�����������%0Returns the value of f evaluated at the point P.T�T�T�%o����������ėȖ���X�%
	__sig__11xZ�Z�%Ȗ�������\�%YReturns the isogeny E0 -> E1 defined by the function field elements phi0, psi0, and omg0.Tata�%ooo||���������ԗȖȗDf�%
	__sig__12 hXh�%ȖЗ����$j�% EllipticIsogeny�k�%��������̗��������� o�EThe divisor d with all occurances of p removed from its factorisation�a�T^^��������,om o8g�  __sig__27�h i� m(o�����j# �  __sig__28Dn\n�n�n �ohp����$True iff J is equal to I����������|p�oppd
__sig__8\t�oxp�����
6/usr/local/magma-2.5/package/AlgGeom/divisors/jacelt.m�a�!6/usr/local/magma-2.5/package/AlgGeom/divisors/jacelt.m�a�"
jacobianlt���%����������̔����%̔��������%)/home/was/magma/Kohel/elliptic_schemes.mg�Y��a��%V)/home/was/magma/Kohel/elliptic_schemes.mg0Z��a��%"Elliptic scheme construction of E.�%�%�%||���������ܖȖЖ�)�%
__sig__0+,$,�% Ȗؖ�����-�% EllipticScheme�/�%��������Ԗ�%"lReturns the elliptic curve E with coefficients given by the sequence S, and assigns the function field of E.6l6�%|���������Ȗ��:�% __sig__1;�;�;�% Ȗ�����=�% __sig__3ield�>�%<Returns the function field of E, followed by the y function.A�A�%|no�������Ȗ��G�% __sig__2GHH�% Ȗ������K�% EllipticFunctionField0M�%�����������%(�Returns a simplified Weierstrass model E1 of the elliptic curve E0, followed by the isomorphisms E0 -> E1 and E1 -> E0 between them.U�U�%|�������� �Ȗ��Y�% Ȗ������[�% WeierstrassModel]x]�]�%����������% The base extension of E to S.\b�%��||��������4�Ȗ(��g�% __sig__4i4iLi�% Ȗ0�����0k�%  BaseExtension�l�%The trace of E at p.o�o�o�o�^^���������om�o��  __sig__35� � � m�o����d � AreLinearlyEquivalent� ����������o�3/usr/local/magma-2.5/package/AlgGeom/divisors/jac.mc�8l��3/usr/local/magma-2.5/package/AlgGeom/divisors/jac.mXc�8l��W���������o�o�o�� __sig__0 d | � �o�o����� � The jacobian of C�!�!�X���������p�o�o�o�$�
__sig__1%�%(&��o�o�����'
__sig__4rveX)IConsider the nonsingular point p as an element of the jacobian of a curve. .Ta������p�op2
__sig__22 383�op�����4�&The degree f unramified extension of L�������������r�r\q�r8
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	__sig__17H�\q�r������%��������(�̔�@�%̔$�������%uu����������4�̔\�% ̔0����� �%  IsLessThan� � �%��������,��%�u��������H�̔#�%  __sig__20�#$�%̔D������%�%��������@��%7Returns the lattice of the r-th externion power module. ,8,X,�%�����������̔T��0�%
	__sig__21�1�1�%̔\�����43�%
ExteriorModule�4�%��������X��%\Returns the exterior product of v with the (r-1)-th exterior power module of L as a lattice.;�;�%�����������x�̔l��>�%
	__sig__22[email protected][email protected]�%̔t�����B�%
ExteriorProduct�D�%��������p��%(�Given a matrix giving a homomorphism V -> W, returns the matrix giving the induced homomorphism of r-th exterior powers of V and W.�M�N�%DD����������̔���Q�%
	__sig__23�S�S�%̔�������U�%
ExteriorMap�VhW�%�����������%6Returns the submatrix of coordinates (i,j) in I1 x I2.]�%DD������������̔��8a�%
	__sig__24pb�b�%̔������e�%
MatrixMinor�f4g�%�����������%6Returns the submatrix of coordinates (i,j) in I1 x I2.m�%
	__sig__25ho�oHackobjCoerceJacCurve7��������pPrint J at level ll;�;����������$p�op�> __sig__3@[email protected][email protected] �o p����B HackobjPrintJacCurveD�D �������� p &The curve containing the elements of JTK S���������p<p�o0pdO �o8p�����P  The affine model underlying J@R W��������Lp�o@p�V __sig__5XLXhX �oHp����Z 1The projective curve containing the elements of Jt]�] X��������\p�oPp�a __sig__6c�c�c �oXp�����e  The zero divisor in the jacobianh�hi a��������lp�opm __sig__7oho�o�o�o�$&}Returns the sequences of sequences of Brandt modules of level N*M, for an order in the algebra ramified at primes dividing N.�$��������x�p�H�d�4?�$
__sig__1�>�$H�l������$$sReturns the sequence of sequences of Brandt modules for the quaternion order defined by QuaternionAlgebra(D1,D2,T).�������������H�t�[email protected]� __sig__2�@� H�|������ PReturns the sequences of sequences of Brandt modules for the quaternion order A.�E������������H���� __sig__3K� H��������M�<Returns the Brandt matrix of theta functions to precision N.P� ����������H���� __sig__4S� H�������@V�  BrandtTheta�@������������4Returns the Gram matrix of the inner product for TM.@��7� ����������H���� __sig__5@��7� H��������@� BrandtGramMatrix ^������������-Returns the matrix of pth coefficients of TM.�b� ��������ГH�ē� __sig__6le� H�̓�����  BrandtHecke���������ȓ� UReturns the Atkin-Lehner operator matrix for the Brandt module BM of quadratic forms.��7� ���������H�ܓ� __sig__7s#g�8�  AtkinLehner� � /home/was/magma/Hecke/hecke.m����  theCategory�+��7� disc�� gammaalp� Z  �  semidecomp� SkF3�� SkZ  �  ModSymBasisl�  BoundaryMap�� old3�� sknew �  sknewdual&'� SkZplus��� SkZminus��  mknewdual� newforms !� oldforms*+� mknew� T_images� X  � mestre'� M./� windinghl� VeCategory� Zdual � qexp%� eigen-� cusplist(T*�  eigenminus� eigenint !� moddeg)�  PeriodLattice3� cinf9�  eigenplus8� cinfimag� Lvals � LRatio'  LRatioOdd01 cp"7 wq>?�  realvolume�C�  imagvolume F iso3� isnew  iscusp% fast-  signspace56 xdata= ZxZalpEFG  qintbasis W  RatPeriodConj RatPeriodMapSign"#  PeriodMap,- PeriodMapPrecision"7K Levelply  Apply g to the ith Manin symbol.MNOP VolLEvenll VolLOddm m  PeriodGens0m PGfast@m  RatPeriodMapHs  RatPeriodLat\sk� decomp�� discQ�  newdecomp�  olddecomp^ 4The generator of the maximal unramified subring of L�_ �����������q\qxq�  __sig__1 � � a \q�q����� � ���������,s\q s��  /home/was/magma/Hecke/hecke.m����G 3Return the level of the space M of modular symbols., D H t��������(u8s u�I  __sig__11��J 4 8suX� �d%1Returns the sequence of subsequences of length r.@e%����������̔��L g% ̔������� h%1Returns the sequence of subsequences of length r.4"\"i%����������̔���%j%  __sig__11�&�&k% ̔�������(l%����������̔�-m%  __sig__12h/�/n% ̔������1o%The factorization of f.p2�2�2p%Q��������Еȕ̔��l6q%  __sig__13�8�8r% ̔ĕ����h:s%"The (rational) factorization of x.�;<t%��������ؕ̔̕|?u%  __sig__14�@�@v% �̔ԕX��Bw%Returns the first n primes.XEpEx%���������̔ܕXKy%  __sig__15�L�Lz% ̔������N{%Returns the first n primes.hP�P|%����������̔�xT}%  __sig__16�U�U~% ̔�����X%  InitialPrimesxY�%����������%Returns the first n primes.�]^�%���������̔�\b�%  __sig__174dLd�% ̔ �����tf�%  PrimesUpToph�h�%����������% Returns the nth prime.m�%  __sig__18lo�o�%  __sig__19�q�qb InertiaElement�c ��������|qd *�The known part of the local element x. Note that this function will be removed in the next release; Expand() should be used in its place.� � e ����������|r�q\q�q<!f __sig__2"4"\"g \q�q�����#h @�The defining polynomial of a generator of L. It is an error if the degree of L is 1. If L is ramified it will be the polynomial defining the ramified extension; otherwise it will be the minimal polynomial of the inertial element-�-i ����������q\q�q�1j __sig__32�2�2k \q�q�����4l 2�The relative degree of L over a substructure. It is an error if the degree of L is 1. If L is ramified it will be the ramification degree, otherwise the degree of inertia�;<m ����������q\q�q|?n __sig__4@�@�@o \q�q�����Bp &The degree f unramified extension of L�Fq �����������q�q\q�q�Kr __sig__5L�LMs \q�q����Ot  LocalRing8PPPu ���������qv >The degree f unramified extension of L, reduced to precision n�Uw �����������q�q\q�q4Zx __sig__6[P[�[y \q�q�����]z JThe totally ramified extension of L defined by the Eisenstein polynomial g�a�a{ �����������q�q\q�qtf| __sig__7hph�h} \q�q����Tj~ "bThe totally ramified extension of L defined by the Eisenstein polynomial g, reduced to precision n�p�p�ptq�  LocalField��� ���������r� >The degree f unramified extension of L, reduced to precision n� � �����������r�r\q�r� �  __sig__18��� \q�r������ JThe totally ramified extension of L defined by the Eisenstein polynomial g,t� �����������r�r\q�r��  __sig__19� � � \q�r����h � bThe totally ramified extension of L defined by the Eisenstein polynomial g, reduced to precision n�#� �����������r�r\q�r|'�  __sig__20<)T)� \q�r���� ,� 3The unramified local ring with prime p and degree fd/|/� ����������r�r\q�r�2�  __sig__21�384� \q�r����x5� @The unramified local ring with prime p, degree f and precision n9�94:� ����������r�r\q�r�<�  __sig__22�=�=� \q�r����,@� SThe totally ramified local ring with prime p defined by the Eisenstein polynomial g�D�D� ���������s�r\q�r�I�  __sig__234LLL� \q�r�����M� cThe totally ramified local ring with prime p and precision n defined by the Eisenstein polynomial g�Q�Q� ���������s s\qsXV�  __sig__24 X4X� \qs����0Z� WThe local ring with prime p and inertia degree f defined by the Eisenstein polynomial g�^�^ _� ���������s s\qs8c�  __sig__25�d e� \qs�����g� dThe local ring with prime p, inertia degree f and precision n defined by the Eisenstein polynomial gl�l�  __sig__26oPo� \q(s�����qL �������� uM 3Return the level of the space M of modular symbols.T l N t��������<�@u8s4up O  __sig__12H  P 8s<u�����Q 8Return the base field of the space M of modular symbols.HtR t����������Pu8sDu<S  __sig__13,DT 8sLu����H U AReturn the Dirichlet character of the space M of modular symbols.P!x!V t��������u8sTuXW  __sig__14�%�%X 4 8s\uX�  'Y  Character))Z ��������Xu� Tnemona-[ SReturn true iff working in the plus one quotient of the space M of modular symbols.�1�1\ t��������xu8slu5]  __sig__15�6�6^ 4 8stuX� �9_ IsPlusQuotient�: ��������pua SReturn true iff working in the plus one quotient of the space M of modular symbols.@,@b t���������u8s�u�Dc  __sig__16�F�Fd 4 8s�uX� He IsMinusQuotientTKf ���������ug Returns the sign of M.�Oh t���������u8s�u�Ri  __sig__17�T�Tj 4 8s�uX� XVk  Returns the isogeny class of A.�X�X�Xl t���������u8s�u\]m  __sig__18�^ _n 4 8s�uX� �o  IsogenyClassaXbp ���������uq WReturn Heilbronn matrices of determinant n, as given by Merel. Here n can be composite.Pjhj�jr ���������u8s�u�os  __sig__19�rdt|t�t) ModularSymbols * ���������t+ [Returns the space of modular symbols of weight k for Gamma_0(N), over the rational numbers.�  , t���������t�t8s�t�- __sig__4��. 8s�t����t/ R'Returns the space of modular symbols of weight k for Gamma_0(N), over the rational numbers. If sign=+1 then returns the +1 quotient, if sign=-1 returns the -1 quotient, and if sign=0 returns the full space. If sign is a positive prime, then the full space of modular symbols mod sign isreturned.X"�"�"0 t���������t�t8s�t�%1 __sig__5&�&�&2 8s�t����)3 ^Returns the the space of modular symbols of weight k for Gamma_0(N), over the finite field Fp. /4 t���������t�t8s�t<25 __sig__63P3h36 8s�t����57 "nReturns the space of modular symbols of weight k for Gamma_1(N) with character eps, over the rational numbers.�:8 �t���������t�t8s�tt=9 __sig__7> ??: 8s�t�����@; "nReturns the space of modular symbols of weight k for Gamma_1(N) with character eps, over the rational numbers.,G< �t��������u�t8s�t|L= __sig__8M�M�M> 8s�t�����O? T4Returns the space of modular symbols of character eps and weight k. The level and base field are specified by the DirichletCharacter eps. The third argument "sign" allows for working in certain quotients. The possible values are -1, 0, or +1, which correspond to the -1 quotient, full space, and +1 quotient.\�\@ �t��������u8s�t�A __sig__9a�aXbB 4 8suX� �dC 4Return the weight of the space M of modular symbols.h�hD t��������u8s u�lE  __sig__10oPoF 4 8suX� �q�rdt^#8����������8st�# 8s|�����(a#"nReturns false if p does NOT divide the discriminant of the subalgeba of Hecke generated by T2, T3, T5, and T7.�b#����������8s���d# 8s�������e# TestDisc�f#�����������#ArtineNMg#/Returns the nth Atkin polynomial modulo p > 2n.(h#����������8s��\i#  __sig__320pj# 8s�������k# AtkinPolynomiall#����������u#  __sig__322mial\*m#"kReturns the pth supersingular polynomial. The roots are the supersingular j-invariants in characteristic p.p1n#������������8s�� 8o#  __sig__32148p# 8s������L8q# SupersingularPolynomial�Cr#����������s#,�Returns a list of the supersingular j-invariants in characteristic p. However, it is *much* faster to use the command SupersingularBasis(Mestre(p)).�Nt#��������؍8s̍PTv# 8sԍ����Ww#6�Given a new factor A=A_f and a prime ell, returns Norm((ell+1-a_ell(f))*(ell+1+a_ell(f))). This is an integer divisible by the congruence primes relating A to a form at level N(A)*ell.f�fx#t���������8s܍ ky#  __sig__323lz# 8s�����,m{#  RibetRaise@m|#���������}#����������8s�s~#  __sig__324�s# 8s�������s�#  SerreModPQ�s�#����������#�����������8s t�#  __sig__3250t�# 8s �����Ht�# SerreModPQTable��{ HeilbronnCremona� | ���������u} MIf n is prime, return Cremona's Heilbronn matrices; otherwise return Merel's.� ~ ��������v8s�u0  __sig__21(@� 4 8s�uX ��  Heilbronn<T� ���������u�  __sig__24GenListToMuot� wtu��������v8s� �  __sig__22!8!� 8sv�����"!%��������d�#%8�Returns the sequence of data consisting of a generator of a Hecke eigenvector, the eigenvector norm, and the eigenvalues of the first t Hecke operators. The content of the cusp space follows.__sig__7-% ��������/&% HeckeEigenvalues1�1�1'%��������p�(%%The Cremona database reference for E.x5)%|����������������d:*% __sig__8;<;T;+% ���������<� �� >�����=,% CremonaReferences@,@-%��������>.%HThe Cremona database reference for linear Hecke eigenspace V of level N.GDG\G/%t ��������Ĕ�����L0%  __sig__10�M�M1% �������� P�( �s/magma/���� ����lar_2%./home/was/magma/Kohel/elementary_arithmetic.mg���3%4./home/was/magma/Kohel/elementary_arithmetic.mg���4%6The representative of x mod n in the range (-n/2,n/2].�Z5%��������ܔ̔Дd_6% __sig__0��7% ̔ؔ�����b8%  BalancedMod�d�d9%��������ԔP% __sig__6heorem�i:%.Returns the Chinese remainder lift of a and b.tl;%LLL�����������̔�dtb%Choose�ut 8s�u�����u HeilbronnMerel�v ���������uw FReturn the Heilbronn matrices of determinant p, as defined by Cremona.� x ���������u8s�u�y  __sig__20(�z 4 8s�uX �� ��������� H������T����������� /home/was/magma/Kohel/decomp.m(��� /home/was/magma/Kohel/decomp.m(���KReturns the sequence of vector subspaces stabilized by the Hecke operators.Lx% ��������������@% __sig__00H% �������L % HeckeDecompositiont � %���������% PReturns the sequence of vector subspaces of V stabilized by the Hecke operators.$$%L%%t �������� ����()% __sig__1*D+�+% �� �����- % __sig__5sition/ % ��������0� �p2 % __sig__3 %  �$�����@�%��������,��\�8:
%HeckeLinearDecomposition;p;�;%��������L�%(�The sequence of one dimensional vector subspaces stabilized by the Hecke operators, priority given to the divisors of the level N.HDD%t��������P�H���<�H%
__sig__3J�JXK%��D�����M%'The sequence of linear Hecke subspaces.�O�O�O%t��������<�X���L�(S%
__sig__4T�T�T%��T������V% Returns the sequence of generators for the linear Hecke-invariant subspaces of the Hecke module.˸__si%
__sig__4��%HeckeThetaDecomposition�%��������(�%8Returns the eigenvalues of Hecke for the first n primes.nvariant su%u��������x��t�%�|��������%%
__sig__6,p %HeckeEigenvectors0uhu�&Unitssors� InitialManinSymbolGenListToSquot0���������v�tu��������(v8s�!�
	__sig__23�"�"�48s$vX $�InitialManinSymbolGenListToM%(&��������� v�t��������<v8s-�8s8v�����/����������v�QReturns a bound b such that T1, ..., Tb generate the Hecke algebra as a Z-module.
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
HeckeBound�����������v�QReturns a bound b such that T1, ..., Tb generate the Hecke algebra as a Z-module.������������v8s�v��
	__sig__31� � �8s�v����"�HReturn vector space with basis the lattice defined by the elements of B.$%H%� t���������v8s�v$)�
	__sig__32@+�+�8s�v����\-�VectorSpaceZBasisd/|/����������v�$sThis function computes the Hecke module generated by a vector v. The result is returned as a subspace with Z-basis. 6h6� utt��������w�v8s�v�:�  __sig__33�;�;� 8s�v�����<�  HeckeSpan$>�>����������v�$uThis function computes the Hecke module generated by a basis of V. The result is returned as a subspace with Z-basis.\G� ttt�������� w8sw�L�  __sig__34�M�M� 8sw���� P�  Compute data needed by FastTn.�Q� t���������(w w8sw�U�  __sig__35�W�W� 4 8swXP tY�  FastTnData�Z�Z� ��������w� tt���������0w8s�a�  __sig__36�c�c� 4 8s,wXX �e� �t ��������Lw<w8sPj�  __sig__37�kDl� 4 8s8wX\ �n� ��������4w�  __sig__38�uv� ��������4v� FastTnis� t ��������vPv8s�6�  __sig__25�8$9�8sLv�����:�

HeckeOperator�;���������Hv�ECompute the n-th Hecke operator Tn on the space M of modular symbols.�@�t��������hv8s\vlE�
	__sig__26DG\G�8sdv�����I�t���������vtv8s�N�
	__sig__27�OP�48spvX
TQ�
TnSparseR�R�R���������lv�"gCompute the action of the Hecke operator defined by the Heilbronn matrices Heil on the sparse vector v.�Z�Z�Z�t���������v8s�vd_�
	__sig__28���48s�vX(
�b�
	__sig__30ond�d�)Return sparse representation of vector v.hTh�u���������v8s�vtl�
	__sig__29�n�n�48s�vX,
pq�SparseRepresentationu�u�,�Compute the characteristic polynomial of the p-th Hecke operator Tn on the space M of modular symbols. Uses the modular algorithm, without proof.
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	__sig__46x��8s�w���������������w�,� Suppose A is a linear transformation of V which leaves the lattice L invariant. Returns A restricted to W, with respect to the basis Basis(W).������������x�w8s�w�
	__sig__47���8s�w����� �$�������� �����#�$
NewtonSlopes%H%�$����������$	__sig__15ctorIndexesT)�$8�Make a list of forms in S_k(N,eps), with eps^2=1, and the correspoding index in the maximal order. The format of the list is [[degrees of eps], isogeny class, disc(O_f), index in max. order].<2T2l2�$����������8���,��5�$ __sig__14�7t8�$��4�����4:�$ QuadraticModularFactorIndexes�;�$��������0��$����������L���$A�$��H�����xC�$SlopesOfUpGamma0ETElE�$��������D��$�����������|M�$ __sig__16$OdO�$��\������P�$
factorpadic�Q�Q�$��������X��$JReturn the p-adic slope alpha part of the polynomial f, to precision prec.�X�X�$��������x���l� ]�$
	__sig__17^�^�$��t�������$
SlopeAlphaPart�a�$��������p��$
	__sig__19a0f4g�$,�Returns the slope alpha part of the characteristic polynomial of T_p acting on S_k(Gamma_0(N)), with p-adic computations done to precision prec.noPo�$��������������,v�$ __sig__18@w�w� ?Compute action of Transpose(Tn) on the Hecke-stable subspace V.P��� �t ��������XwTw8sHw � 4 8sPwX � � t� ��������hww8s@�  __sig__39�� 8s\w����T� ?Compute action of Transpose(Tn) on the Hecke-stable subspace V.���� tt �������xwpw8sdw� �  __sig__40� � � 4 8slwXh "� ?Compute action of Transpose(Tn) on the Hecke-stable subspace V.�$�$�$�tt��������w�w8stw�(�
	__sig__41�*�*�8s|w�����,�LCompute action of Transpose(Tn), n in plist, on the Hecke-stable subspace V.01�tt��������w�w8s�wh4�
	__sig__425x5�8s�w����t8�LCompute action of Transpose(Tn), n in nlist, on the Hecke-stable subspace V.;�;�tt��������w8s�w�>�
	__sig__43[email protected][email protected]�8s�w����B�@�Compute the action of the Atkin-Lehner involution Wq on M, when this makes sense (i.e., trivial character, even weight). The Atkin-Lehner map Wq is rescaled so that it is an involution, except when k>2 and char(F) divides q.M�M�M�t���������}�w8s�w�Q�
	__sig__44�R$S� 8s�w����0U� Wq(VXV� ���������w4 SkbolsZ� ,�Compute the conjugation involution * on V. This involution is defined by the 2x2 matrix [-1,0,0,1]; it sends X^i*Y^j{u,v} to (-1)^j*X^i*Y^j {-u,-v}.apa� t ���������w8s�w@f�  __sig__45 hTh� 8s�w���� j� StarInvolution�k� ���������w� t���������w8s�w,v� Tnpoly w" DCompute the action of T_p on a subspace of integral modular symbols.P# t ���������x8s�x( b$CoefficientValuationpc$��������|�l$__sig__8ionPrimeToP@d$dReturns the valuation of the ideal generated by the coefficients a_n of f with n not divisible by p.te$#���������������g$��������h h$CoefficientValuationPrimeToP � i$����������j$/Cast g, which should be in Z[[q]], to lie in R.�$%H%k$#$#��������������$)m$���������+n$
CastPowerSeries\-o$����������p$KTruncate f so that q^prec is the highest possible power of q involved in f.343q$##��������̑����7r$
__sig__99<9�9s$��ȑ�����:t$TruncatePowerSeries�;�;u$��������đv$"jReturns a sequence of tuples <[N,iso,dim],[cp1,cp2,...],[wp1,wp2,...]>, one for each new factor of J_0(N).@CxCw$������������ؑ�Gx$
	__sig__100JTJy$��������Lz$
ComponentGroups�M{$��������ܑ|$ cReturns a sequence of triples <dim,[cp1,cp2,...],[wp1,wp2,...]>, one for each new factor of J_0(N).U0U}$���������������tY~$
	__sig__11�Z�Z$�������� ]�$
	__sig__13nt^�^�$��������������b�$
	__sig__12�d�d�$�������4g�$ComponentGroupsPrintiHi�$����������$*�Returns the sequences of slopes of the Newton polygon of f with at the prime p. The slopes are the valuations of the p-adic roots of f.vv,v�$�� ������w�wDx� Restrict�  ���������w ,� Suppose A is a linear transformation of V which leaves the subspace W invariant. Returns A restricted to W, with respect to the basis Basis(W). H   t ��������xx8sx�  __sig__48�� 8s x���� ] Suppose A leaves W invariant. Returns A restricted to W, with respect to the basis Basis(W).�  8 ��������(x x8sx8!  __sig__490"X" 8s x�����# 2� Suppose A is a linear transformation of V which leaves the subspace W spanned by the vectors listed in B invariant. Returns A restricted to W, with respect to the basis B.+�+  ��������0x8s$x�/
	__sig__5041L18s,x�����2
>�Suppose B is a basis for an n-dimensional subspace of some ambient space and A is an nxn matrix. Then A defines a linear transformation of the space spanned by B. This function returns the kernel of that transformation.�;�;t��������Lx@x8s4x$?  __sig__51�@�@ 4 8s<xX� �B KernelOnD�D�D ��������8x t t��������Tx8s�L  __sig__52�M�M 4 8sPxX�  P ^Compute the action of the Hecke operator Tn with respect to the basis for the subspace V of M.tT tt ��������dx8sXxhX  __sig__53Z0Z 8sx�����[ HeckeOperatorOn�] ��������\x 9Compute the action of Tn on the integral modular symbols.�cd t ��������|x8spxi  __sig__548jPj 8sxx����tl HeckeOperatorMkZnoPo! ��������tx$
	__sig__55v,v%8s�x�����w9$��������Ԑ:$PGiven a space M of modular symbols and a field F, attempt to compute M tensor F.

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;$tt���������8s�0 =$48s�Xh>$ /home/was/magma/Hecke/apps.m0��:�?$,/home/was/magma/Hecke/apps.m1��:�@$#������������A$
__sig__0��B$�������� C$
	OrderList��D$���������E$ZTwists the q-expansion f by the Dirichlet character eps, thus returning sum eps(n)*a_n(f).�%�%F$##�������� �����)G$
__sig__1+, ,H$�� ������-J$����������$fcpableesK$0�Compute the Zp-rational newforms associated to f. The precision of Zp is of course not infinite; it is actually O(p^20), which could cause problems. Be careful!8�8$9L$��������D�8���,��;M$__sig__2<�<�<N$��4������>O$ZpRationalNewforms�@�@P$��������0�Q$t������T�L����GR$
__sig__3I0JTJS$��H������LT$4Returns a list of the the Zp-rational newforms in M.O�OU$t��������\���P�$SV$__sig__4T�T�TW$��X������VX$?Returns the power series sum_{n not divisible by p} a_n(f) q^n.ZtZ�ZY$##��������l����_Z$__sig__5t�[$��h�����Xb\$pDeprived0dHd]$��������d�f$__sig__7ioniHi^$FReturns the valuation of the ideal generated by the coefficients of f.m_$#������������x�du$
__sig__6v@vTva$��������Dx& HeckeOperatorSkZH' ���������x( .�Given a cusp a=[u,v] this function finds the index i and a scalar alpha such that a = alpha*(i-th standard cusp). It then returns the tupple <alpha, i>. � � ) t���������x8s�x�*  __sig__56<T+ 4 8s�xX� �,  CuspToFreel�- ���������x. Compute the Boundary map.�  / tw���������x8s�x�"0  __sig__57�#�#1 4 8s�xX� �%3 tt���������x8s2  __sig__605 8s�x�����-< ���������x6 CuspidalSymbols<27 ���������x8 <Return the subspace of S_k(N,Q) of cuspidal modular symbols.7t89 tt���������x8s�x�;:  __sig__59t<�<; 4 8s�xX� �=C  __sig__62mbols�?= tt���������x8s�D>  __sig__61�E�F? 8s�x�����GQ ����������Ly8s�$��������<�8s0�d$8s8������ $Volumed
$��������4�$BCompute the norm of an element of a quotient of a polynomial ring.@X$o����������T�8sH�0$

__sig__354�$8sP�����l $Returns the word for nth.� � $��������d�8sX��#$

__sig__355$%L%$8s������&$ToNth�'$��������\�$#The sequence of primes [p1 ... p2].�-�-$��������|�8sp��1$ __sig__356�2�2$8sx������4 $PrimeSeq5�5�5!$��������t�I$Twistter"$-Base extend the character eps to the field F.0<#$������������8s���?$$ __sig__357A(A% 4 8s��XP|C& BaseExtendCharacterXEpE'����������(& Base extend the mlist to the field F.PL)������������8s�� P*  __sig__358Q(Q+ 4 8s��XT�R, BaseExtendMlist�T-����������.% Base extend the quot to the field F.�Y/����������Đ8s���]0  __sig__359�_�_1 4 8s��XXta2 BaseExtendQuot<c3����������<  __sig__361tionXh4/Base extend the decomposition D to the field F.0kTklk5��������ܐ8sА�p6  __sig__3600uhu7 8sؐ�����v8 BaseExtendDecomposition�x�%�@ IntegralCuspidalSymbolsPLA ���������xB tt��������y8s�QD 8s y���� TE SkZ U4UF ��������yG tt��������y8s�[H  __sig__63x]�]I 8s y�����_T DegeneracyCosetReps� U ��������DyV EIf possible compute the map M1 --> M2 corresponding to the integer d. W tt ��������tydy8sXy� X  __sig__66�Y 8sy����Z  DegeneracyMap[ ��������\y\ ZIIf possible compute the degeneracy map M1 --> M2 corresponding to the integer d. Here M1 and M2 are spaces of modular symbols, of the same weight and d is a divisor of the quotient of the two levels N1 and N2. If the N1 divides N2 then the "raising" map is computed, if N1 is divisible by N2, then the "lowering" map is computed.�%&] tt ��������|y8spy*^  __sig__678,X,_ 4 8sxyX  . VReturns the space Mk(NN,eps') associated to Mk(N,eps). Here NN must be a divisor of N.<2a tt���������y8s�yx5b  __sig__6877c 4 8s�yX�9d OldModularSymbols ;;e ���������yf  Returns the p-new subspace of M.> ??g tt���������y8s�yxCh  __sig__69E<Ei 4 8s�yX\Gj  pNewSubspaceHIk ���������yl %Returns the p-new subspace of M dual.dOm tt���������y8s�y@Rn  __sig__70 TtTo 4 8s�yX Vp pNewDualSubspaceX X4Xq ���������yr tt���������y8s _s  __sig__71t�t 4 8s�yX Xbv ���������y� MkZnSymbolw tt���������y8shkx  __sig__72�lmy 4 8s�yX�oz CuspidalNewSubspace|t�t{ ���������y|  Returns the p-new subspace of M.xyy(y<yJ SkZPluspaK �������� yL tt��������8y8sHiM  __sig__64hj�jN 8s4y�����lO SkZMinusnoPoP ��������0yu SknewcesR  __sig__65�v wS 4 8sHyX�xyy�  Mknewdual��� ��������z� tt��������z8s �  __sig__75X p � 4 8s zX0� �  Sknewdual��� �������� z�  __sig__77space� tt��������8z8s��  __sig__76��� 4 8s4zX4� �#@Create a table of odd parts of BSD data for N between N1 and N2.!�!�!�#����������8���8s����# TableCongruenceR&l&�&�#�������� ��#>Append the level N of the congruence table to the file "name".\-�#����������Џ8sďL1�#  __sig__347T2l2�# 8s̏����84�#�����������܏8s�8�#  __sig__348�94:�# 8s؏����T;�# TableConnected\<�#��������ԏ�#*�Create a table of connectedness of abelian intersection graph in the "box" T_k(Gamma_0(N)), where N1 <= N <= N2 (N prime) and k1 <= k <= k2.ElE�#�����������8s�TK�#  __sig__349�L�L�# 8s������N�#���������8s�Q�#  __sig__350�S�S 8s�������UReturns the odd part of n.�WX�����������8s��[  __sig__351t]�] 4 8s �X8�_��������������������8s4g  __sig__352�hi 8s ������j DotProdDl �������� � *�Compute the absolute value of the determinant of the matrix got by finding a Z-lattice basis for the Z-span of the rows in the sequence L.�x�x  __sig__353�y�y} tt���������y8s�y ~  __sig__73� 8  4 8s�yX(� �  NewSubspace� � � ���������y� tt��������z8st�  __sig__74�t� 4 8s zX,\�#(Create a table of factored characteristic polynomials of Tp on S_k(Gamma_0(N)) for N between N1 and N2 and p between p1 and p2.� � � �#�����������8s�� �# 8s ������ �#  TableCharpoly�#����������#'Create a table of congruence numbers r.for N betwee�#����������ȏ�8s ����#  __sig__346�j��# 8s(�����8s�# TableBSD"4"\"�#�����������#7Append the level N of the BSD table to the file "name".�&�&�&�#����������@�8s4�,�#  __sig__339�-�-�# 8s<������/�#@Create a table of odd parts of BSD data for N between N1 and N2.2�2�2�#�����������P�8sD�l6�#  __sig__340�8�8�# 8sL�����h:�#  TableBSDFull;X;�#��������H��#7Append the level N of the BSD table to the file "name".[email protected][email protected][email protected]�#����������h�8s\��D�#  __sig__341�FG�# 8sd������H�#BCreate a table of rational parts of special values of L-functions.hM�M�#����������x�8sl�(Q�#  __sig__342RDR�# 8st������T�#  TableLRatio�UV�#��������p��#  __sig__345sZdZ�#'Create a table of splittings of J_0(p). ]H]]�#��������������8s��ta�#  __sig__343c<c�# 8s������Pe�# TablePrimeSplittingsg�g�#�����������#0Append the level p of the prime splitting table.mDn\n�#������������8s���u�#  __sig__344�v�v�# 8s�������x�# 8s�������y� CuspidalNewDualSubspace� � ��������0z� #Returns the new subspace of M dual.�!�!� tt��������Pz8sDz�� 4 8sLzX8�&� NewDualSubspace�'� ��������Hz� 8�Computes the image in M2 of x in M1 under the theta operator q*d/dq : M1 --> M2. Thus M1 and M2 should be spaces of modular symbols mod ell of the same level, and Weight(M2) = Weight(M1)+ell+1.T3l3� uttu��������hz8s\zd7�  __sig__78�9�9� 8sdz�����:�#��������|�8s��# 8sx������ �#  ArtinOrderTwo� �#��������t��#�� Given an integer N that is coprime to 5; a sequence [d1,...,dr] of degrees which define a mod N character eps; and a sequence [p1,p2,...,pn] of primes p such that a_p=0; this function computes the subspace ker(Tp1') meet ker(Tp2') ... meet ker(Tpn') of the space M_2(Gamma_1(5*N),eps_5*eps;F5bar)^+. (Primes denote transpose.) The first argument returned is the subspace W, and the second is the space of modular symbols. The subspace W is a subspace of the dual and can be computed efficiently using FastTp.�#�#tt��������8s���&�#  __sig__331�(�(�# 8s�������+�#����������OddPart/�#"hReturns the modular symbols factor corresponding to the nth curve from the paper "Empirical evidence..."3�3�3�#t������������8s��t8�#  __sig__332�9�9�# 8s������;�#  EvidenceCurve,<�#�����������#4�Returns the modular symbols factor corresponding to the curve from the paper "Empirical evidence..." labeled s. Here s is given by the notation in that paper. E.g., "29", or "65,A".,G�#t��������Ď8s��|L�#  __sig__333�M�M�# 8s�������O�#t��������Ў8sS�#  __sig__334�T�T�# 8s̎�����V�#  EvidenceLabelhX�#��������Ȏ�#7Create a table of eigneforms on Gamma_0(N) of weight k.^^�^�#�������������8s܎�b�#  __sig__335�d�d�# 8s�����4g�# TableEigenformsi�#����������#EAppend the level N data for the table of eigenforms to the file name.(p�#�����������8s��v�#  __sig__336hx�x�# 8s�������y�#  __sig__337lz�z�  ThetaOperator\<� ��������z� #Returns the winding element {0,oo}.�@�@� tu���������z�z8stzlE�  __sig__79DG\G� 8s|z�����I� WindingElementLL� ��������xz� &Returns the winding submodule T{0,oo}.�P� tu���������z�z8s�z�T�  __sig__80�UV� 8s�z����4X� WindingSubmoduleYZ0Z� ���������z� 9Returns the winding element X^(i-1)*Y^(k-2-(i-1))*{0,oo}.D\� tu���������z8s�z�d�  __sig__81Xfpf� 8s�z����i� <Returns the submodule Te spanned by the ith winding element.ltl� tt���������z8s�zdt�  __sig__82�uv� 8s�z�����w� t���������z�z8sTz�  __sig__96��� 4 8s�{X���  ToIsogenyCode � ���������{  __sig__100isor � 2Returns the number n so that s = ToIsogenyCode(n).� � � ���������{8s�{(�  __sig__97l�� 8s�{�����"��������ċ8s�� �" 8s������ �" LinearRelations� �"�����������"��������؋8s&�"  __sig__299�& '�" 8sԋ����T)�" MonomialsOfDegreeD, ,�"��������Ћ�"���������8s<2�"  __sig__300P3h3�" 8s�����5�" RelationsOfDegreeD�67�"����������"���������8s�<�"  __sig__301�=�=# 8s������,@# OCompute the dimension of the space of cusp forms of character eps and weight k.�D�D�D#��������� ��8s��I#  __sig__3024LLL# 8s ������M#  DimensionSk�O�O#���������#KCompute the dimension of the space of cusp forms on Gamma_0(N) of weight k.XU�U#��������(�8s ��Y #  __sig__303[([ # 8s�����\] # Returns dim S_k(Gamma_1(N))._�_ #��������8�8s,�d #  __sig__304de�e# 8s4�����Th#  DimensionSkG1�i#��������0�#��������L�8sdt#  __sig__305�uv# 8sH������w#  DimensionS2�x�x#��������D�1#TpSSularJ� &xGiven a Manin symbol [P(X,Y),[u,v]], this function computes the corresponding element of the space M of Modular symbols. � 8 �  __sig__89H  � 4 8s@{Xx� �  __sig__93bol� � -Returns the modular symbol {alpha,beta} in M.�� tu��������d{T{8sH{��  __sig__90<\� 8sP{���� � ConvFromModularSymbol � ��������L{� HGiven a modular symbol P(X,Y)*{alp,beta}, FromModularSymbol returns the corresponding element of M. The input is a pair <P,x>, where P is a polynomial in X and Y, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.)�)�)� tu��������t{l{8s{ /�  __sig__9140l0� 4 8sh{X��1� NGiven a sequence of modular symbols P(X,Y)*{alp,beta}, FromModularSymbol returns the corresponding element of M. The input is a sequence of pairs <P,x>, where P is a polynomial in X and Y, and x=[[a,b],[c,d]] is a pair of elements of P^1(Q), where [a,b] <--> a/b and [c,d] <--> c/d.t<�<� tu��������|{8sp{[email protected]�  __sig__92�A�A� 4 8sx{X��D� RTests the modular symbols conversion routines. Must always return the basis for M.�HI� t�����������{8s�{�M� 8s�{���� P� TestConvP QQ� ���������{�  __sig__95bols0U� =Return the sub Z-module M_k(N,Z) of integral modular symbols.�X� tt���������{8s�{ ]�  __sig__94^�^� 8s�{������ IntegralModularSymbolsXb� ���������{� tt���������{8s j� 8s�{����Dl� IsogenyCodeToInteger� ���������{� �������� |�{8s� �  __sig__98� � � 4 8s�{X�� SmallestPrimeNondivisor ���������{ KReturn the smallest prime number ell not dividing N and such that ell >= p.x� ��������|8s||   __sig__99t �  4 8s|X�|! Returns the nth prime.�" ��������|8s|(& 4 8s |X��' NthPrime)@)X) �������� | 1Returns the integer n so that p is the nth prime.�/0 ��������L|<|8s0|83  __sig__101�4�4 4 8s8|X�l6 PrimePos8�8�8 ��������4| HReturns the position of the divisor p of N, or 0 if p does not divide N.==x= ��������T|8sH|�A  __sig__102DC|C 8sP|����pE Fast, non-proved, charpoly.�G�G  ��������d|8sX|0M  __sig__103�NO 4 8s|X��P ModularCharpoly�Q ��������\|  ��������x|8slX  __sig__104 Z4Z 4 8st|X��[ FactorCharpoly�] ��������p|! >Compute integer kernel of the not-necessarily-square matrix A. d" w8���������|�|8s�| i#  __sig__105<jTj 8s�|����xl%  IntegerKernel�n& ���������|' >Compute integer kernel of the not-necessarily-square matrix A.�w( D8���������|�|8s�|Xz)  __sig__106�{�{�{�{� ���������{Y IsNewnns� FReturns the n-th isogeny coding. The coding goes A,B,C,...,Z,AA,BB,...�x� ���������{8s�{{�"����������"H�Let eps_p denote the character obtained by restricting eps to the p-factor. This function returns eps_p(n). When gcd(p,n)>1 this is 0. When gcd(p,n)=1, let m be an element of (Z/NZ)^* which is 1 mod q for all q neq p and is n at p. Then eps_p(n) = eps(m).���"����������8�8s,���" 8s4������"�����������D�8s �"  __sig__292� � �" 4 8s@�X|� �"NReturns true iff M>=1 divides Modulus(eps) and is divisible by Conductor(eps).�#�"���������T�8sH� '�"  __sig__293)()�" 4 8sP�X��+�"  CanReduceModM-�"��������L��"  __sig__296ersP1�"*�Given a Dirichlet character eps, such that M divides Level(eps) and Conductor(eps) divides M, return the associated mod M Dirichlet character.7�"����������l�8s�(;�"  __sig__294<0<�" 4 8sh�X�x=�" AssociatedModMCharacter|?�"��������d��"GReturns a sequence consisting of all of the mod N Dirichlet characters.�F�FG�"����������8sx�PL�"  __sig__295hM�M�" 8s�������O�" AllDirichletCharacters�P�"��������|��"  __sig__297erClassesU4U�" aReturns a sequence consisting of the Gal(Qbar/Q)-conjugacy classes of mod N Dirichlet characters.LZdZ�"����������8s���^�" 8s��������" AllDirichletCharacterClassesb�b�"�����������" fPrint a Dirichlet character. This is a temporary function, while I wait for classes or "hack objects".lk�"�������������8s���p�" 8s�������u�".�Let P be a sequence of power series (or polynomials) over the integers. This function returns a basis for the Z-linear relations satisfied by these series.�z�z�"  __sig__298�{|* 8s�|�����+ 'Compute integer kernel of the matrix A. ( @ ,  8���������|8s�|D -  __sig__107 4 . 8s�|����l/ ECompute integer kernel of the matrix A, and return as integral space.x0  8���������|8s�|@1  __sig__1080H2 8s�|����L 3 IntegerKernelZ 4 ���������|5 UCompute the linear combinations of the elements of B defined by the elements of Comb.�6 ���������|8s�|�(7  __sig__109�*�*8 4 8s�|X�,�" Create a Dirichlet character modulo N. The input zeta is a root of unity in some field. The second input "degs" is a list of nonnegative integers. We explain the notation we use for degs. There is an isomorphism Z/NZ ---> (Z/p1^(i1)Z) x (Z/(p2)^(i2)Z) x ... with p1 < p2 < ... the primes dividing N. Passing to units we find that each factor in the right hand side is a cyclic group, except possibly when p1=2 in which case the cyclic group is trivial, or a group of order 2 times a cyclic group (possibly trivial). The degs specify the order that the image of a primitive root of each Z/p^iZ should map to. The primitive root used is the one computed using the function PrimitiveRoot(p^i). In the special case p=2 and i>=3, one should supply supply two entries corresponding to 2 in the degs sequence, because the group (Z/2^nZ)^* is generated by -1 and 5. The entries in the gens sequence correspond bijectively, and in order, to the prime divisors of N, listed in increasing order. Note: Not every character can be described in this way. /L/�"���������Ȋ��8s��l2�" 4 8s��XX84�"'Create the trivial Dirichlet character.h6�6�6�"���������؊Њ8sĊ�:�"  __sig__285�;�;�" 8s̊����,=�" ]Create the trivial Dirichlet character with values in the prime field of characteristic p>=0.�A�"����������8sԊ�F�"  __sig__286�GH�" 8s܊�����K�" ^Evaluate the character eps on the matrix gamma=[a,b, c,d] in Gamma_0(N). The result is eps(a). P�"�������������8s��S�"  __sig__287U0U�" 4 8s�XhhW�",Evaluate the character eps at the integer n.Z0Z�"������������8s�^�"  __sig__288�_,�" 4 8s��Xl�a�"4Evaluate the character eps at the integer (n mod N).eLe�"L�����������8s� j�"  __sig__289�k�k�" 4 8s �XpXn�",Evaluate the character eps at the integer n.rdt�"L����������0� �8s�Dx�"  __sig__290(y<y�" 8s �����Tz�"  EvaluateAtp�{�{�"  __sig__291�|�|�|�|9 LinearCombinations�/�/: ���������|; t t��������}�|8s5<  __sig__110�6�6= 8s�|�����9> IntegerKernelOn�:? ���������|@ F�Suppose B is a basis for an n-dimensional subspace of some ambient space and A is an nxn matrix. Then A defines a linear transformation of the space spanned by B. This function returns the integer kernel of that transformation, as a subspace with basis.\GA  t�������� }8s}�LB  __sig__111�M�MC 8s}���� PD  Compute the lattice index [V:W].Q�Q�QE ���������� }8s}VF  __sig__112�WXG 8s}�����YH  LatticeIndex[([I ��������}J "gReturns true if and only if A is a root in the ModSymParent tree, i.e., if and only if A has no parent.8c�c�cK t��������4}8s(}�hL  __sig__113�i jM 4 8s0}X DlN  ModSymIsRootnXnO ��������,}P 3Returns the parent of the modular symbols factor A.�v�vQ tt��������L}8s@}zR  __sig__114�z{S 4 8sH}X||�|�|^  OldFactor��_ ��������l} (List the dimensions of the factors in D.  � a ���������}8s�}0 b  __sig__1170c 8s�}�����d  List the Shas of factors in D.te ��������؄�}8s�}<f  __sig__118,Dg 8s�}����H h Sha� � i ���������}~ J0forms"j 6Returns the number of newforms in the decomposition D.�k ���������}8s�}t(l  __sig__119�)*m 8s�}�����,�"(�Using the standard vector std, this function creates a Dirichlet character modulo N that takes values in a field of characteristic p.l2�"��������������8s|��5�" 8s�������8�" DirichletCharacterL:d:�"�����������"(Create any Dirichlet character modulo N in characteristic p. The input "pows" is a list of nonnegative integers. There is an isomorphism Z/NZ ---> (Z/p1^(i1)Z) x (Z/(p2)^(i2)Z) x ... with p1 < p2 < ... the primes dividing N. Passing to units we find that the right hand side is a cyclic groups, except possibly when p1=2 in which case the cyclic group is trivial, or a group of order 2 times a cyclic group (possibly trivial). Let zeta be a primitive rth root of unity in the algebraic closure of the prime field of characteristic p. The pows specify the power of zeta that a primitive root (computed using PrimitiveRoot(p^i)) of each Z/p^iZ maps to. The primitive root used is the one computed using the function PrimitiveRoot(p^i). In the special case p1=2 and i1>=3, the user must supply two entries in the pows sequence. (But if i1=1 or 2, then the user must supply exactly one entry.) The group (Z/2^nZ)^* is generated by -1 and 5. The entries in the pows sequence correspond bijectively, and in order, to the prime divisors of N, listed in increasing order. i0iHi�"�������������8s���m�"  __sig__282�o�o�" 4 8s��XL�t�"�������������8s���x�"  __sig__283Pyhy�" 4 8s��XT�z�"Create Dirichlet character.(|@|�"  __sig__284P}d}T  ModSymParent U ��������D}V t��������}8s� W  __sig__115� � X 4 8s\}X(� Z ��������X}[ tt��������t}8s�\  __sig__116<\] 8sp}���� � -Returns the sum of all cuspidal factors of M. � tt��������T~8sH~� � 8sP~����� F"  __sig__263� � G" 8s������ H" ClassicalPeriodI"���������J"@Returns the classical period r_j(f) = int_{0}^{ioo} f(z) z^j dz.@K"tV��������4�8s(�L L"  __sig__264� M" 8s0�����<!N"NCompute the G_1 function defined in section 2.13 of Cremona's Algorithms book. O"VV��������D�8s8��'P"  __sig__265@)X)Q" 8s@�����,"����������8s��p0�" 4 8s�X0�1�" CharacterLevel�2�"�����������"0 Return true iff the Dirichlet character is odd.89(9�"��������� �8s�<�"  __sig__276�<=�" 8s������>�"F Returns the field in which the Dirichlet character eps takes values. �B�"���������,�8s �G�"  __sig__277�HI�" 4 8s(�X8PL�"  FieldOfValues�M�"����������"  __sig__279magesQ�"���������@�8s4U�"  __sig__278pV�V�" 4 8s<�X<�X�"DirichletCharacterImagesZ�Z�Z�"��������8��"-Returns the standard vector that defines eps.��"���������X�8sL� e�" 8sT������g�" StandardVectorLi�"��������P��";Returns the orders of the images of the factors (Z/p^iZ)^*.p,p�"���������p�8sd��v�"  __sig__280lx�x�" 8sl������y�"  ImageOrderspz�z�"��������h��"  __sig__281|}�}�}�}n NumberOfNewforms � � o ���������}x  __sig__121tors "p >Returns the number of cuspidal factors in the decomposition D.�q ���������}8s�}x(r  __sig__120�)*s 8s�}�����,t NumberOfCuspidalFactors/u ���������}v 0List the Atkin-Lehner signs of the factors in D.3T3l3w ���������}8s�}d7y 8s�}�����9z BCompute J_0(N). (Really this is a decomposition of H1(X_0(N),Z)^+.H<<{ ����������}8s�}[email protected]|  __sig__122tA�A} 8s�}����D� FullCuspidalFactor� � ��������L~� 9Compute the Hecke decomposition of the old subspace of M. ( � t��������l~8s~� �  __sig__128��� 8sh~������ OldDecomposition�� ��������d~� ;Compute the Hecke decomposition of the *NEW* subspace of M.��� t��������x��~8sx~� �  __sig__129""� 4 8s�~Xtx#� ;Compute the Hecke decomposition of the *NEW* subspace of M.<&T&� t����������~8s�~�*�  __sig__130x,�,� 4 8s�~Xx /� NewDecomposition0�0�0� ���������~� %Compute the Hecke decomposition of M.�4� t����������~8s�~�9�  __sig__131|:�:� 8s�~�����;� FullDecomposition�<�<� ���������~� tt���������~8s\D�  __sig__132TElE� 8s�~�����G� ���������~� t���������~8s�O�  __sig__133�P�P� 4 8s�~X��Q�  IsCuspidal�ST� ���������~� WqOnWX� t���������~8s�[�  __sig__134t]�]� 4 8s�~X��_�  IsEisenstein8a� ���������~� :Compute the action of the Atkin-Lehner involution Wq on A. hTh� t ��������8s�~tl�  __sig__135�n�n� 8s�~����pq� ���������~� 5Returns the sign of the Atkin-Lehner involution on A.�x� t��������8s �{�  __sig__136�|�|� 8s����~ ���������}� GetVtion� #Compute the SemiDecomposition of M.dL|L� t��������� ~ ~8s~LP�  __sig__123<QTQ� 8s~����S� SemiDecompositionU0U� ��������~� #Compute the SemiDecomposition of M.HZZ� t���������,~~8s~�^�  __sig__124D\� 8s ~�����a� ACompute the SemiDecomposition of the subspace Vdual inside Mdual.Xfpf� tt���������<~4~8s(~,k�  __sig__125�l�l� 8s0~�����o� SCompute the SemiDecomposition of the subspace Vdual inside Mdual, starting with Tp.xv�v� tt���������D~8s8~�y�  __sig__126�z�z� 8s@~����@|�  __sig__127P}d}�  DecompFast��� ��������8� ����������T8s �  __sig__139X p � 4 8sPX�� � ��������8s�  __sig__140�� 4 8s\X�t�  LabelHelper�<� ��������X� Printx � OLabel the factors in the decomposition D, then return the sorted decomposition.�!""� ��������x8slH%�  __sig__141l&�&� 4 8stX�t()"t������������8s\-+" 8s������|/," EllipticInvariants�0 1-"����������.".Computes w1, w2, c4, c6, and j to precision n.H5/"t��������Ĉ8s��4:0"  __sig__258 ;;1" 8s������\<2",Returns the abelian variety associated to A.>�>3"t��������Ԉ8sȈ�B4"  __sig__259�D�D5" 8sЈ����,G6" AbelianVariety�H7"��������̈8"HComputes an elliptic curve associated to the weight-two rational modular symbols factor A (this curve is well-defined up to isogeny). By the Shimura-Taniyama theorem the computed curve is correct (assuming correctness of our implementation of the algorithms).�VhW9"t���������8s�([:"  __sig__260] ];" 8s����� _<"tV�����������8s8c="  __sig__261�d e>" 8s������g?"  LAnalytic0iHi@"���������A"&yCompute L(A,j) for j a "critical integer", i.e., one of the integers 1,...,k-1. Use at least n terms of the q-expansions.�u�uB"tV���������8s�yC"  __sig__262�yzD" 8s �����X{E"tV��������,� �8sp~X"DerivG20� SignWN � ��������� ����������,8s� �  __sig__137� � � 8s(����� � DecompWq��� ��������� t����������L@8s\�  __sig__138,� 4 8s<X�� / *�Give the subspace A of the DUAL of M, attach necessary data to make it a factor of M. For this to make sense, A should be Hecke stable.� � < 0 ttt����������Ȁ8s�� 2 8sĀ����4@ DecompZ3 t ��������Ԁ8s�4  __sig__160��5 4 8sЀX�6  DecompFastTp�7 ��������̀8 t���������8s�"9  __sig__161�#�#: 8s�����L%; DecompCharpolyTp&�&�&< ���������= tt����������8s .>  __sig__162�/0? 8s�������1 " 8s0������2"t��������P�@�8s�6"  __sig__2519(9" 8s<������:"  PeriodLattice�;"��������8�" VComputes the complex period lattice associated to A using n terms of the q-expansions.�A"t��������X�8sL��F"  __sig__252�G�G" 8sT�����XK"tV��������t�d�8shO"  __sig__253hP�P" 8s������Q"  RealVolume�S�S"��������\� ".�Computes the volume of A(R), R the RealField. This function returns the volume of the identity component times the number RealTamagawa(A) of real components.�] "tV��������|�8sp��a "  __sig__254�c�c " 8sx������e "tV������������8sTj!"  __sig__255�kHl"" 8s�������n#" ImaginaryVolume�p"����������%"8Computes the volume of A(iR), the pure imaginary points.x�x�x&"tV����������8s���{'"  __sig__256�|�|(" 8s������~*"  __sig__257 4�  LabelFactors � � ��������p� t������������8sL%�  __sig__142p&�&� 4 8s�X�x(� ���������� Print the factors in D.�-�-�-� �������������8s��1�  __sig__143�2�2� 4 8s�X��4 newforms� 8Return the number of new factors of the decomposition D.9�9�9� ���������8s��<�  __sig__144�=�=� 8s������? NumNewformFactors�  �������� � t��������(�8s�  __sig__149� � 8s������ �������� �# Factor D�Returns the modular factor "Nk[Weight][IsogenyClass]", where N is the level, k is the weight, and [IsogenyClass] is a letter such as "A", "B", etc. An alternative input format is "N[IsogenyClass]" in which case the weight is assumed to be two.� �  t��������L�@�8s4�H#  __sig__150 X 8s<�����&  ModularFactor ' ��������8� t��������X�T�8sL/  __sig__151�0�0 8sP����� 2 t��������d��8sH5  __sig__152�67 8s\������9 tt��������t�l�8s\<  __sig__153\=t= 8sh�����x? >�Returns the index into the D list of the new factor with the indicated isogeny class. If the iso classes have not yet been computed yet, they are computed. If there is no factor of class iso this function returns 0.TJ�JTK t��������|�8sp�dO  __sig__154dP|P 8sx������Q tt������������8sV!  __sig__155�WX" 8s�������Y ����������% tt������������8s�&  __sig__156�a�a' 8s������Hd( t������������8sHi)  __sig__157hj�j* 8s�������l+ >�Returns the index into the D list of the new factor with the indicated isogeny class. If the iso classes have not yet been computed yet, they are computed. If there is no factor of class iso this function returns 0.<yPyhy, t������������8s��@|-  __sig__158P}d}. 8s�������~1  __sig__159���  NumOldFactors�A� ���������� .Return the indexes of new factors in the list.�G� ����������8s�,M�  __sig__145�N O� 8s�����|P�  NewFactors�Q�Q� ���������� .Return the indexes of new factors in the list.hW� ����������8s�([�  __sig__146] ]� 8s����� _� NewformFactors�� ���������� 8Return the number of new factors of the decomposition D.g�g�g� ���������8s��k�  __sig__147�m�m� 8s�����(p  NumNewFactors�t ��������� 8Return the number of new factors of the decomposition D.y�y�y ���������8s�||  __sig__148x}�} 8s�����K  CuspOrder��L �������� �M eCompute the order in A of the difference (x)-(y) of the cusps defined by the 2-tuples alpha and beta. N t����������@�8s4��O  __sig__165��P 8s<�����t� LRatiotionQ qComputes the following upper bound on the torsion subgroup of A: gcd { #A(F_p) : 3 <= p <= 19, p not dividing N }� � R t�������P�8sD�8!S  __sig__1660"X"T 8sL������#U  TorsionBound�V ��������H�W sComputes the following upper bound on the torsion subgroup of A: gcd { #A(F_p) : 3 <= p <= maxp, p not dividing N }�-�-X t��������h�8s\��1Y  __sig__167�2�2Z 8sd�����h4c  __sig__169dMapx5[ bhCompute the rational period mapping. The period mapping is scaled so that the integral modular symbols SkZ are taken surjectively onto the lattice Z^d. Note: the choice of rational period mapping is well-defined only once A is created, and can be different if A is created again; i.e., it is with respect to the non-canonical basis used internally to define A.D�D�D\ tw��������x�8sl�I]  __sig__168�KL^ 8st�����|M_ ScaledRationalPeriodMap�O ��������p�a ICompute the order in A of the cusp defined by X^i*Y^{k-2-i}*{alpha,beta}.XU�Ub t����������8s���Yd 8s������|[e t8����������8s,f  __sig__170Papag 8s�������ch  ModularKernelLei ����������j B�Compute the modular degree of A. This is the degree of the canonical map from the dual of A to A. When A corresponds to an elliptic curve, this is the SQUARE of what is frequently referred to as the "modular degree" in the literature.xv�vk t8����������8s���yl  __sig__171�z�zm 8s������@|n  ModularDegreed}o ����������r  __sig__172�,�A ���������B tt���������8s� C  __sig__163x � D 8s ������ E  DecompZdual� � F ���������G 2Compute the order in A of the difference (0)-(oo).<TH t��������8�(�8s �,I  __sig__164��J 8s�����h � 9Return the level M of the character group of X_0(pM)/F_p.��� ����������8s��� �  __sig__186� � 8s�����4 � XGroup_M 4� ����������� >�Compute the n-th Hecke operator on the toric part (+Eisenstein series) of the closed fiber of the Neron model of J_0(pM) over F_p. The second argument returned is a diagonal matrix which is double the weight vector.| � � � � �������� �8s� "�  __sig__1874#L#� 8s������i!uu������������8s��x(k! 8s�������*m!����������|!  __sig__223iety�/n!;Returns the dimension of the abelian variety attached to A.(2@2o!t��������ą8s��|5p!  __sig__221 7d7q! 4 8s��X �9r! DimensionAbelianVariety(;s!����������t!fxAttempt to compute the q-expansion of one of the Galois conjugate modular forms associated to A. This function uses theta series, so if successful it can be used to compute a very large number of Fourier coefficients quickly. If there is no prime which exactly divides Level(A), Weight(A) gt 2, the character of A is nontrivial, then the function qEigenform(A,prec) is called.MN�Nu!t#��������܅8sЅ�Qv!  __sig__222�S�Sw! 8s؅�����Ux! qEigenformThetalWy!��������ԅz!5Returns the at least 7 terms of the q-expansion of A. ]{!t#����������8s�<a}! 8s�����<c~!  qEigenform�d e!����������! fReturns the q-expansion of one of the eigenforms associated to A, computed to precision at least prec.�m�!t#�������� �8s��u�!  __sig__224|v�v�! 4 8s�X4�xR"DerivG1ts�!*�Returns generators for the saturation of the Z-module generated by the sequence of q-expansions. Uses the smallest degree as precision.�}�}�}�!�������� �8s��^"DerivG3ԁp .�Returns the congruence number r = [S_k(N,Z) : (This) + (Complement)], along with the structure of the quotient; q-expansions are computed to precision prec.%�%q t8������́8s���)s 8sȁ����\,t  CongruenceR�-�-u ��������āv 8�Computes the invariants of the group-theoretic intersection of the abelian varieties corresponding to the modular factors A and B. If the factors are not distinct, then the 0 group is returned.9(9w tt8����������8s؁<x  __sig__173�<=y 8s������>� B�For an integer s in the critical strip (1<=s<=k-1), return the quotient L_A(s)*(s-1)! / (2pi)^(s-1)*Omega, which is a rational number. Here Omega is the volume of A(R), if s is odd, and the volume of the -1 eigenspace for conjugation.� � � t��������P�8sD���  __sig__178(�� 8sL������� BdEDing� TReturns the odd part of the rational part of L(A,s); potentially faster than LRatio. � � t��������l��8sT�8!�  __sig__1790"X"� 8s\������#�  LRatioOddPart�� ��������X�� t��������t�8s ,�  __sig__180�-�-� 8sp������/�  Returns true iff L(A,eps,1) = 0.1�1�1� �t����������8sx�5�  __sig__181�6�6� 8s�������9� LEpsilonVanishes:�:�:� ��������|�� <Returns the degrees d dividing p-1 such that L(A,eps,1) = 0.?x?� t����������8s��\D�  __sig__182TElE� 8s�������G� LEpsilonVanishing0JTJ� ����������� TM�M�N� Z[25\� precS��X�� "iReturn the character group X of the toric part of the closed fiber of the Neron model of J_0(p) over F_p.�UV� ���������ԂĂ8s��Z�  __sig__183�[�[� 8s�������]� XGroupd_� ����������� .�Return the character group X of the toric part of the closed fiber of the Neron model of J_0(pM) over F_p. [RIGHT NOW RETURNS THE EISENSTEIN PART AS WELL.]�i j� ���������t�܂8sЂPo�  __sig__184�ppq� 8s؂�����u� BReturn the characteristic p of the character group of X_0(pM)/F_p.�x�x� ����������8s��{�  __sig__185�|�|� 8s�����~� XGroup_p 0� ���������� TXn��z AbelianIntersectionpA�A{ ��������܁| LReturns the group-theoretic intersection of a sequence S of modular factors.HI} 8����������8s��M~  __sig__174�O�O 8s������Q� 8�Compute "[Phi(SkZ^+):Phi(W)]". If Plus is false, compute instead "[Phi(SkZ^-):Phi(W)]". It should be the case that W tensor Q is contained in SkZ^+ tensor Q (or SkZ^- tensor Q, when Plus is false).0Z� tt�������� �8s�^�  __sig__175�_,� 8s������a�  PeriodIndex�c�c� ���������� PReturns the lattice spanned by the basis of W under the period map defined by A.kPkhk� tt�����������8s��p�  __sig__176,udu� 8s ������v�  PeriodImage�wDx� �������� �� t��������H�8�8s�|�  __sig__177�}�}� 8s4�����0� ��������0��  PhiX_and_mX��� ��������D�� /Returns the dimension of the character group X.� � � � ���������d�8sX� �  __sig__190Ph� 8s�������  XDimension�(� ��������\�� 6�Returns the character group of the toric part of the closed fiber at p of the space M of modular symbols. This only makes sense when p exactly divides the level of M and M has weight two.� � � t�����������|�8sp��#�  __sig__191 %H%� 8sx������&� *�Returns the character group corresponding to the largest prime p which exactly divides the level. If no such p exists, an error results.-. .� t�����������8s�� 2�  __sig__192 343� 8s�������4� =Compute XGroup for xmallest prime exactly dividing the level.9� tt������������8s���;�  __sig__193�<�<� 8s�������>� ����������� (�Return the factor of the character group corresponding to the p-adic rigid analytic optimal and co-optimal quotient associated to A.GH� t8����������8s��|M�  __sig__194OdO� 8s�������P� ]Compute the p-adic modular degree of A, at the largest prime which exactly divides the level.0U� t��������ԃă8s��tY�  __sig__195�Z�Z� 8s������ ]� PhiX]^� ����������� fCompute the order of the image of the component group of J_0(N) in the component group of A, all at p.pf� t��������܃8sЃ,k�  __sig__196�l�l� 8s؃�����ol!Dotceor� YCompute the p-adic modular degree of A, at largest prime which exactly divides the level.�wDx� t�����������8s��z�  __sig__197�{|� 8s�����d}� XModularDegree�~� ���������� 'Compute the p-adic modular degree of A.<�d���� ���������� XGroupV�  Returns the monodromy weights.� � ���������P�4�8s(�� �  __sig__188� � � 8s0������ MonodromyWeights�� ��������,�� 4Computes the quantities PhiX and mX associated to V.�� 8������� �L�8s@�� �  __sig__189� � � 8sH�����">!.�Let T be an nxn matrix over K with irreducible characteristic polynomial f. This function returns an eigenvector for T over the extension field K[x]/(f(x)). � @!  __sig__212� � A! 4 8s�X� B!8�The elements of gens should all lie in a field Q[x]/(f(x)). It is assumed that they generate a subring O of Q[x]/(f(x)), as a *Z-module*! This function returns the discriminant of this Z-module.��C!���������8s�� D!  __sig__2134 l E! 8s������ F! ZAlgDisc!�!�!G!�������� �H!*�If A is newform modular factor, compute the discriminant of the ring generated by the Fourier coefficients of one of the q-expansions of A.�)�)I!t��������,�8s �/J!  __sig__21480p0K! 8s(������1L! DiscriminantOf�2M!���������N!*�If A is newform modular factor, compute the discriminant of the ring generated by the Fourier coefficients of one of the q-expansions of A.p;�;O!t��������D�8s8��>P!  __sig__215[email protected][email protected]Q! 8s@�����BR! DiscriminantKf�DS!��������<�T!KCompute the motive B made from the *=sign subspace of the modular motive A.hL�LU!tt��������\�8sP�PPV!  __sig__216@QXQW! 8sX�����(SX! SignedSubspace�TY!��������T�Z!+Compute eigenvector for sign subspace of A.LZdZ[!tu��������|�t�8sh��^\!  __sig__217H]! 4 8sp�X�a^!pReturns an eigenvector of the Hecke algebra on A over a polynomial extension of the ground field. A must be new.h�h i_!tu����������8sx�m!  __sig__218lo�oa! 4 8s��Xhtb! ZReturns the images of the ith standard basis vector under the Hecke operators Tp for p<=n.�x�xc!t����������8s��\{d!  __sig__219l|�|e! 4 8s��X�}f!  HeckeImages�~g!����������h!#Compute the dot product of v and w.��̂j!  __sig__220�,�� t���������8s��X&!  __sig__198@'�'! 8s������)!+Computes the quantity PhiX associated to V.--!88�������8s�P1!  __sig__199X2p2! 8s�����<4! SReturns the Mestre module of M. The level must be prime and the weight must be two.9(9!t8���������8s�<!  __sig__200�<= ! 8s ������> !  MestreGroup[email protected][email protected] !�������� � !4�Return the factor of the character group corresponding (maybe) to the p-adic rigid analytic optimal quotient associated to A. The representation is via the Mestre construction.L�LM !t8��������<�8s0��P!  __sig__201�Q�Q! 8s8����� T !LLet v be a choice of eigenvector, corresponding to A, on the module of supersingular points. This function returns sum w_i (v_i)^n, a quantity which, by Gross-Zagier, is relevant to the computation of the order of vanishing of L(A,s) at 1. Currently Conductor(A) must be prime.� !!t������������8s���#! 8s�������!(�Compute the order of the group of *geometric* points of the component group at the largest prime that exactly divides the level. 0 H %!t������������8s���!&!  __sig__206�"�"'! 8s������(! ComponentGroup�%)!����������*!:�Compute the order of the group of *geometric* points of the component group at p, assuming that p exactly divides the level. If working in the +1 quotient, then only the odd part of the order is returned.1�1+!t����������8s���4,!  __sig__207 6h6-! 8s������9.!:�Compute the order of the group of Fp rational points of the component group of A at the largest prime which exactly divides the level of A. WARNING: Stein has not yet nailed down the power of 2 when Wp=+1![email protected]/!t��������ȄĄ8s���D0!  __sig__208�F�F1! 8s�������H2!t��������Є8s�M3!  __sig__209�O�O4! 8s̄�����P5!��Try to compute the odd part of the order of Sha. Assuming BSD we obtain a number which is either zero or divides the true order of Sha, and only misses primes where the representations are reducible. The following caveats apply: (a) We assume the BSD formula. (b) We use the upper bound on torsion coming from Hecke operators, so the result of this function might be too small. (C) At primes whose square divide the level we do not know how to compute cp; however such cp can only be divisible by p and primes dividing the order of the torsin group. We simply make our conjectural value of Sha coprime to such primes when there is a p whose square divides the level.@nXn6!t���������8sԄ�u7!  __sig__210�v�v8! 8s܄�����x9! u�����������8sX{:!  __sig__211h|||;! 8s������}<!  Eigenvector�~=!���������?!  u��������l��8s��(�@�T�!  MestreGroupVVXV!��������4�!t����������\�P�8s\]!  __sig__202�^ _! 8sL������! MestreEigenvectorpb�b!��������H�!88����������d�8sPj!  __sig__203�kDl! 8s������n!6�This function returns sum w_i (e_i)^n, a quantity which, by Gross-Zagier, is relevant to the computation of the order of vanishing of L(A,s) at 1. Currently Conductor(A) must be prime.yPyhy!u8������������t�8sh�@| !  __sig__204P}d} ! 8sp������~ ! MestrePowerSum� !��������l�"!  __sig__205���! SaturatePolySeq��!����������!,�Returns the index and structure of the Z-module generated by the sequence of q-expansions in its saturation. Uses the smallest degree as precision. 0 �!������4�8s(���!  __sig__226Ht�! 8s0�������! IndexInSaturationt��!��������,��!  __sig__228ntegralBasis� �!MReturns the linear combination of qIntegralBasis(A) which gives Eigenform(A)."�!tt��������L�8s@�H%�!  __sig__227l&�&�! 8sH�����t(�!#tV��������ć��8s��\-�! 8s������|/�! FastPeriodIntegral�0 1�!�����������!#tV��������̇8s�6�!  __sig__245�89�! 8sȇ�����:�!<�Given a homogeneous polynomial P(X,Y) of degree k-2, a 2x2 matrix g=[a,b,c,d], and a q-expansion f of a weight k modular form, this function returns the period <f, P {oo,g(oo)}> = Int_{oo,g(oo)} P(z,1) f(z) dz.HB�B�!#tV��������܇8sЇ\G�!  __sig__246�HI�! 8s؇����LL�! SlowPeriodIntegral�M�M�!��������ԇ�!t�������8sT�!  __sig__247XU�U�! 8s������W�! PeriodGeneratorsXHYtY�!����������!t��������,��8s\"  __sig__248�a�a" 8s�����d"  PeriodMapping�e"����������"(Apply the period mapping Phi to v in Mk.k�k�k"uu�������� �8s�pq"  __sig__249|u�u" 8s������v" ApplyPeriodMapping�x�x "��������� ":�Computes the complex period mapping associated to A using n terms of the q-expansions. The period map is a homomorphism Phi:Mk(N,eps;Q)--> C^d, where d=dim A. Furthermore, A(C) := C^d/Phi(Sk(N,eps;Z)).؀� "t��������4�8s(�(� "  __sig__250����!  __sig__225� �! 8s�����P�!  __sig__235���! 8s�������!  __sig__240gationg�!3Returns the map r_A : M_k(Q) ----> M_k(Q)/Ker(Phi).L d �!t���������8s�� �!  __sig__236��! 8s�������! RationalPeriodMappingX�!�����������!EReturns the map r_A : M_k(Q) ----> M_k(Q)/(Ker(Phi)+(sign quotient)). �!t���������8s �|!�!  __sig__237�"�"�! 8s������#�!SignedRationalPeriodMapping�%�%�!����������!8Returns a basis for the image of S_k(Z) in M_k/Ker(Phi).,<,\,�!t��������0�8s��0�!  __sig__238�1�1�! 8s,�����83�! RationalPeriodLattice�4�!��������(��! XReturns matrix of "conjugation" with respect to the basis RationalPeriodLatticeBasis(A).;�;�;�!t ��������H�8s<�(?�!  __sig__239�@�@�! 8sD������B�!RationalPeriodConjugationE@E�!��������@��!NComputes the number of real components of the abelian variety associated to A.0M�!t���������8sT��P�! 8s\�����DR�! ImaginaryTamagawa�T�T�!��������X��!NComputes the number of real components of the abelian variety associated to A.,[�!t��������x�8sl��_�!  __sig__241�<a�! 8st�����<c�!  RealTamagawad e�!��������p��!EComputes <f, {alpha,oo}> for alpha any point in the upper half plane.Hl�!V#V������������8s���q�!  __sig__242�u�u�! 8s������w�! PeriodIntegral�x�!�����������!FComputes <f, P{alpha,oo}> for alpha any point in the upper half plane.�|�!V#V����������8s����!  __sig__243���! 8s���������!Compute <f, {oo, g(oo)}>.D�X��!  __sig__244��̅�! EigenformInTermsOfIntegralBasis� �!��������D��!JReturn an integral basis for the sum of the spaces in the decomposition D.���!��������t�d�8sX�X�! 8s�������! qIntegralBasis��!��������\��! SReturn an integral basis for the sum of the cuspidal spaces in the decomposition D.T!|!�!����������|�8sp�\�!  __sig__229�%�%�! 8sx����� '�!t������������8s\,�!  __sig__230�-�-�! 8s������0�!B�Compute an integral basis for the space spanned by the Galois conjugates associated to A. If A is a space of modular symbols, returns an integral basis for the corresponding space of cusp forms. The base field must be the rationals.9�9�9�!t����������8s���<�!  __sig__231�=�=�! 8s�������?�!/Compute the sum of two modular symbols factors.�B�B�B�!ttt����������8s���G�!  __sig__232�I�I�! 8s�������L�!  AddFactors�M�N�!�����������!<�Compute the intersection of two modular symbols factors. If the intersection is zero-dimensional use the function "Intersection" to compute the structure of the corresponding intersection of abelian varieties.Z0Z�!ttt����������8s��^�!  __sig__233�_,�! 8s�������a�! IntersectFactorsc�cd�!�����������!&~Given a list polyprimes of pairs <p, f(x)> computes the following subspace of the dual of M: intersect ker(f(Transpose(Tp))). �m�!tt���������؆8s̆�u�!  __sig__234xv�v�! 8sԆ�����x�!  DualSubspaceyhy�!��������І�!<� Given a sequence polyprimes [<p1, f1(x)>,...<pr,fr(x)>] of pairs a prime and a polynomial, this function computes the following subspace of the subspace W of the dual of M: intersect ker(f(Transpose(Tp))). Ё���!ttt���������8s���S"��������<�T"NCompute the G_2 function defined in section 2.13 of Cremona's Algorithms book.� U"VV��������\�8sP�0V"  __sig__266(@W" 8sX������Y"��������T�Z"NCompute the G_3 function defined in section 2.13 of Cremona's Algorithms book.|["VV��������t�8sh�h \"  __sig__267�!�!]" 8sp�����#_"��������l�"6�Compute (d!)*Prod L^(dr)(fi,1) over the d Galois conjugates of f. TODO: When both r > 1 and d>1 the normalization is wrong! It's trivial combinatoric to work out but I have not done it.�-�-a"#V����������8s���1b"  __sig__268�2�2c" 8s�������4d" DerivWt25�5�5e"����������6#T5]f"tV������������8s�<g"  __sig__269>�>h" 8s������[email protected]i"tV������������8s�Dj"  __sig__270�F�Fk" 8s�������Hl"8�Return the leading coefficient of Taylor expansion about the critical integer j and the analytic rank of the the L-series associated to A. The precision used is at least n terms of the q-expansion.�Qm"tV��������8s��Vn"  __sig__271�WXo" 8s�������Yx"  __sig__273ield([p"4�Returns two arguments, a v-adic field K and an element zeta in K that is a primitive pth root of unity. Here v should be a prime number, or 0; if v=0 then K=the complex numbers.�cdq"����������̉8s��ir"  __sig__2728jPjs" 8sȉ����tlt" CompleteCyclotomicField�tu"��������ĉv"> Returns true iff the character eps is the trivial character. �yw"����������8s؉�|y" 4 8s�X(�}z") Returns the level of the character eps. ��{"����������8s� �|"  __sig__274D�X�}" 4 8s�X,��~") Returns the level of the character eps. �� ��"  __sig__275L�d�|����(p1list�(ne.mmm�(Sgensecform�(SquotDf�(Scoef�k�(Tgens �(Tquot�s�(��H����ģ�H����(  GetCategory,{خ�( <x x x  x0�(M���( ˸�(أ0ԣУܣ(�,�H����(  SetCategorygory;�( |P|| |D�(��(̣ȣ�(Meturn�( categoryx����(�(�4� ��@�H����(  NormalizePair//////�( �����(-Bad x=%o, N=%o, u=%o, minv=%o, s=%o, mint=%o �h�(N, u, minv,)�� ���� � ��(�,�0�4�8�<�)x����)Z�4<x)uPl�X) v1];rRing();x) a)R7�)N8r)g Z ! u;0,)s )dЯl  )minv )mint�� )Ng�p@s )vNg00)t*Ng;;r)kPl�X),W)L�lP�H�h�H���) P1Reduce, m0Q�hQ�)�0 <  L  \  ���  �   �����         (  8 D   ���� p  |    �     �� ��� �    �  d < H  X    0)T�X�\��d�)x//////////)list) N1Reduce(x, list))nrn 0, 1;)sPl�X)�� )  EnumerateP1 cL�>#8���������8s�t~?#  __sig__312Hh@# 8s�������A#KComputes Tp on the Mestre graphs module. p must be a prime between 2 and 7.�,�B#8 ���������8s�h�C#  __sig__313l���D# 8s �����H�� SlopeAlphaPartGamma0 � ������������4Return the smallest valuation of a coefficient of f.���������������� ��������0� SlopeValuation ������������  __sig__21ngruences����������������#�  __sig__20%L%� ���������&�pAdicFamilySlopeCongruences�1�1����������������������В��t8� ��̒����4:� pAdicFamilyTable;<;T;���������Ȓ�#/home/was/magma/Startup/shortcuts.m�'��0��#/home/was/magma/Startup/shortcuts.m0(��0��Fast, non-proved, charpoly.�D�D� �����������I� __sig__0K�KL� ������|M�  modcharpoly�O�O����������� factor(charpoly(f));S�S� ������������X� __sig__1XLYxY� ������,[����������� factormod, as in PARI0��������� ���Ld� __sig__2e�eDf� � ������h�  factormod�ij����������� /home/was/magma/Startup/custom.mo�:�@?�<% __sig__1�==% ̔�����>%.Returns the Chinese remainder lift of a and b.?%rrL�������� ��̔��<?@% __sig__2�>A% ̔�����B%.Returns the Chinese remainder lift of a and b.7C%LrL�������� ��̔��D% __sig__3�t6E% ̔�������7F%.Returns the Chinese remainder lift of a and b.G%rLL��������,��̔�H% __sig__4KI% ̔ ������KJ%1Returns the Chinese remainder lift of the list A.�7K%�LL��������4�̔(���7L% __sig__5,RM% ̔0�����O%��������D�̔8��@Q% ̔@�����R%  IsSquareFree@��7S%��������<�T%+Returns the quotient by the largest square.�@U%��������\�̔P�V% __sig__7�@��7W% ̔X�����X% SquareFreeRoot7Y%��������T�Z%;Returns the sequence of subsequences of [1..n] of length r.Pd[%����������t�̔h�\% __sig__8f]% ̔p������h^%3Returns the sequence of subsets of r elements of S._%��������������̔x���7% __sig__94sa% ̔������c%��������|�f%  __sig__10� /home/was/magma/Startup/custom.m�S�xW��'ab�JThe twist of E over the quadratic extension obtained by adjoining Sqrt(d).�]�||��������@�0�4�� __sig__0 b� 0�<������'/home/was/magma/Kohel/brandt_modules.mgxW��'/home/was/magma/Kohel/brandt_modules.mgxW��(�Returns the sequences of sequences of Brandt modules of level N, for an maximal order in the algebra ramified at primes dividing N.�i���������h�X�H�L�� __sig__0p� H�T�����4s�  BrandtModules0s���������P��( � 4�t(�( \� �(  EchelonBasis�?�(��������h��(2/home/was/magma/Kohel/supersingular_polynomials.mg h��p��( 2/home/was/magma/Kohel/supersingular_polynomials.mgph��p��( __sig__3nomials@�()Returns the gth supersingular polynomial.�I�(�����������������( __sig__0 L�( ���������(.Returns the first g supersingular polynomials.�@�(�����������������( __sig__1�@��7�( ���������(SupersingularPolynomialsV�(�����������(5Returns the first g supersingular polynomials over R.�@�(�����������������( __sig__2�@��7�( ���������(BReturns the puiseux series expansion of the kth Eisenstein series.��7�(����������Ȣ�����@�( ��Ģ�����( EisensteinSeries��7�(�����������(AReturns the Petersson inner product of modular functions f and g.k�(�����������������Ԣ�( __sig__4Xq�( ��ܢ�����( PeterssonInnerProduct���7�(��������آ(  LeftOrderForm�(���������(AReturns the Gram matrix of the norm form of the right order of I. ( (� ��������(�|� �� (  __sig__19�� ( |������� ( RightOrderForm� (�������� � ( OReturns the Gram matrix of the norm form of basis B for the quaternion ideal I.� �  (� ��������@�|�4�x!(  __sig__20�"�"( |�<������#(  BasisForm %H%(��������8�(+Returns the left ideal of A generated by M.�*�*(����������X�|�L�|/(  __sig__21�0�0( |�T�����<2(  LeftIdealP3h3(��������P�(,Returns the right ideal of A generated by M.9�9(����������p�|�d�,<(  __sig__22=,= ( |�l�����? (  RightIdeal�@�@ (��������h� (!Returns the conjugate ideal of I.�F�F (������������|�|�L!(  __sig__23M,M"( |�������dO#( IdealConjugate|P(����������%("kReturns the composite I*J in A, where I and J are respectively right and left ideals in the same algebra A. X4X&(�������������|��� \'(  __sig__24�]�](( |�������,)( IdealCompositepa*(����������+( _Returns the composite Conjugate(I)*J in A, where I and J are left ideals in the same algebra A.�i�i j,(�������������|���Po-(  __sig__25�ppq.( |��������u/( LeftSkewComposite�v w0(����������1( Returns the composite I*Conjugate(J) in A, where I and J are right ideals in the same algebra A.|�|�|2(�����������Р|�Ġ�3(  __sig__26؀�4( |�̠����<�5( RightSkewComposite��ȃ6(��������Ƞ7(CReturns the ideal of A generated by elements of the form x*y - y*x.��؈8(�����������|�ܠ��9(  __sig__27����:( |������Ԏ;( Returns the suborder Z + P in A.���Ȑ<(�������������|����=(  __sig__28�H�>( |�������?( aReturns an element of A whose coordinates in terms of the reduced basis are random elements of I.���@(�����������|���<�(@Returns the Gram matrix of the norm form of the left order of I.  ( (� ���������|��� (  __sig__18��( |� �������' |��������' Returns the reduced trace of x.t���'�����������8�|�,�� �' __sig__8 0 h �' |�4������ �'  ReducedTrace!�!�'��������0��' Returns the reduced norm of x.�%�'�����������P�|�D�T)�' __sig__9+�+�+�' |�L������-�'  ReducedNormd/|/�'��������H��'  __sig__11nomial2�'JReturns the characteristic polynomial of the quaternion algebra element x. 6h6�'�����������h�|�\��:�'  __sig__10�;�;�' |�d������<�'AReturns the conjugate of x as an element of a quaternion algebra.�@�@�'����������x�|�l�<E�' |�t�����\G�'@Returns the bilinear product of x and y in a quaternion algebra.LdL|L�'��������������|�|�LP�'  __sig__12<QTQ�' |�������S�' BilinearProduct�T�'�����������' PReturns true if the quadratic form of B1 is smaller than that of B2, else false.[�[�[�'���������ȡ��|���\�'  __sig__13�a�a�' |�������d�',Returns the commutator x*y - y*x of x and y.g�g�'�������������|����k�'  __sig__14�m�m�' |�������(p�'  Commutator|t�t�'�����������'CReturns the Gram matrix of the norm form of a quaternion algebra R.�y�y�'� ��������ȟ|����|�'  __sig__15�}�}�' |�ğ����0�' NormForm��,��'�����������' _Returns the Gram matrix of the norm form of the quaternion ideal I, followed by the ideal norm.8�P�h��'� ���������|�ԟ���'  __sig__16X�p��' |�ܟ�������'  IdealFormč��'��������؟�' _Returns the Gram matrix of the norm form of the quaternion ideal I, followed by the ideal norm.Ԓ� ��'� ����������|����'  __sig__17@�X��' |���������'  ReducedForm���(����������'o||��������l����X"�' ��h������#�' VeluImageCurve��'��������d��',/home/was/magma/Kohel/quaternion_algebras.mg������'F,/home/was/magma/Kohel/quaternion_algebras.mg������'  inclusion_map|/�'  reduced_basis�0�':Returns the inclusion map of I in its left or right order.P3h3�'�����������|���7�' __sig__09�9�9�' |��������:�' Returns the left order of I.<\<�'������������|���,@�' __sig__1ApA�A�' |�������\D�'  IdealOrderTElE�'�����������')Returns the reduced basis of the ideal I.dL|L�'�����������|���LP�' __sig__2Q<QTQ�' |�������S�'  ReducedBasisT�T�'�����������'B�Returns the associative algebra over the integers defined by generators x and y, where Z[x] and Z[y] are quadratic suborders of discriminant D1 and D2, respectively, and Z[x*y - 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