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Author: William A. Stein
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	__sig__23��k(���������l(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.@)X)m(�����������̡������X.n(
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	__sig__93�<�<�)��X������>�)KeepPGroupWeights�?�?�)��������T��)5Check the PC presentation of G for the Hall property.lE�)�$������t���h�<H�)
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	__sig__31	0P	�P	�(��H������T	�(@�Calculates the split extension lift for all modules stored in SQP. The return are sequences for each prime of: -1 iff the solution space does not exist in general, otherwise it is the Fq-dimension of the space (could be 0).�	@�	ة	�(�R��������\���P�(�	�(
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__sig__1h��P#t#������4/usr/local/magma/package/Geometry/Crv/plane_curves.m*	H,	�4/usr/local/magma/package/Geometry/Crv/plane_curves.m*	H,	�.�Construct the plane curve, of type Crv, in the same ambient space, given input any plane curve of special type: CrvRat, CrvCon, CrvEll, CrvHyp, or CrvMod.�x�ss���������#�#�#�"�<�A curve W in Weierstrass form isomorphic to the nonsingular genus 1 curve C; the second return value is a map of the ambient space restricting to an isomorphism C -> W taking the point p to the the flex at infinity�yssx������@##4#��
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HermitianCurve|����������#�0/usr/local/magma/package/Geometry/CrvRat/autos.m8	�9	�0/usr/local/magma/package/Geometry/CrvRat/autos.mP8	�9	�$oThe automorphism taking the three element indexed set of points S to the three element indexed set of points T.������Hv��������l$�#�#�#D#d#�#	$pParametrization of the conic curve C; if provided, the projective line P will be used as the domain of this map.
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JacHypHeight'�'"����������,|,�,,NGiven a function f such that f(z, n) = F(z) + O(p^n), where F is a power series with coefficients in the p-adic integers: F(z) = f_0 + f_1 z + f_2 z^2 + ... with Valuation(f_m) >= d*m, where d is non-negative, this function finds the coefficients f_m up to O(p^n), for m up to k. �+,����[ScaledIgusaInvariants�\��������`-]0�Compute the Igusa J-invariants of a polynomial of degree at most 6, scaled by [16, 16^2, 16^3, 16^4, 16^5]. The coefficient ring must not have characteristic 2.h�^8R���������-0-t-_
__sig__3,`0-|-�����a.�Compute the Igusa J-invariants of the curve y^2 + h*y - f = 0. The polynomial h must have degree at most 3, and the polynomial f must have degree at most 6.�b88R���������-�-0-�-<"d0-�-�����#e
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__sig__0$|,�,����g8�Compute the Igusa J-invariants of a polynomial of degree at most 6. The integer 2 must be a unit of the coefficient ring, and if Quick, the base field must not be of characteristic 2, 3, or 5.�
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__sig__5:|,�,����;2�Given a sequence of points P_i on a Jacobian (of a curve of genus two over the rationals), this returns the matrix (<P_i, P_j>), where < , > is the canonical height pairing. <������������,|,�,=
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__sig__9J|,$-����K//usr/local/magma/package/Geometry/CrvG2/igusa.m��	L //usr/local/magma/package/Geometry/CrvG2/igusa.m��	�DoublepM,�Given a polynomial of degree at most 6, compute the Clebsch invariants A, B, C, D (as on p. 317 of Mestre) in characteristic other than 2, 3, or 5.N8R��������P-@-0-4-O
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__sig__4antsWF�Compute the Igusa J-invariants of the curve y^2 + h*y - f = 0, scaled by [16, 16^2, 16^3, 16^4, 16^5]. The polynomial h must have degree at most 3, the polynomial f must have degree at most 6, and the characteristic of the base ring should not be 2.X88R��������x-h-0-\-Y
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JInvariants��t���������-u8�Compute the Igusa J-invariants of a polynomial of degree at most 6. The integer 2 must be a unit of the coefficient ring, and if Quick, the base field must not be of characteristic 2, 3, or 5.4Lv8R���������-�-0-�-�w
__sig__8$<x0-�-�����y(�Compute the Igusa J-invariants of a genus 2 curve over a field. If Quick, the base field must not be of characteristic 2, 3, or 5.4zRR���������-0-�-|"{
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	__sig__13ants�(~8R��������..0-�-D-l-�-r0-�-�����x�
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__sig__0�� 242������/The hyperelliptic curve given by y^2 + h y = f.���8��R��������P2H2 2<2��
__sig__1� 2D2����(�/The hyperelliptic curve given by y^2 + h y = f.�����8R��������`2X2 2L2��
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� 2T2����T$�)The hyperelliptic curve given by y^2 = f.2�nopqrst}B�Compute the Igusa-Clebsch invariants A', B', C', D' of a polynomial of degree at most 6 (see p. 319 of Mestre). The integer 2 must be a unit of the coefficient ring, and if Quick, the base field must not be of characteristic 2, 3, or 5.
	__sig__10�0-�-�����IgusaClebschInvariants����������-�<�Compute the Igusa-Clebsch invariants of the curve y^2 + h*y - f = 0. The polynomial h must have degree at most 3, and the polynomial f must have degree at most 6. These will be all be zero in characteristic 2.�88R�������� ..0-.�
	__sig__11�0-.�����4�Compute the Igusa J-invariants of a genus 2 curve over a field. These will be all be zero in characteristic 2, and if Quick, the base field must not be of characteristic 2, 3, or 5.�RR��������(.0-.�
	__sig__12�0-$.�����<�Compute the 10 absolute invariants as on p. 325 by Mestre: J_2^5/J_10, J_2^3*J4/J_10, J_2^2*J_6/J_10, J_2*J_8/J_10, J_4*J_6/J_10, J_4*J_8^2/J_10^2, J_6^2*J_8/J_10^2, J_6^5/J_10^3, J_6*J_8^3/J_10^3, J_8^5/J_10^4�RR��������8.0-,.�0-4.�������������X/� eGiven the images on the Kummer surface of points P, Q, P-Q on the Jacobian, returns the image of P+Q.�������������x/x.l/P�
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	__sig__14�*�*�x.�/�����-�,0/usr/local/magma/package/Geometry/CrvG2/kummer.m�	�	�x.�.�����8The point on K specified by the sequence of coordinates.8�����$��������.�.x.�.l�x.�.������!The origin on K, where zero is 0.����$��������.�.x.�.��
__sig__3��x.�.�����The image of P on K.���$�������d4�.x.�.��
__sig__4��x.�.������&zCheck if point specified by the homogeneous coordinates s is on K. If so, return corresponding SrfKumPt as a second value.D-P�wCompute the 2-Selmer group of J (a Jacobian of a hyperelliptic curve defined over the rationals). The first value is the upper bound for the Mordell-Weil rank deduced from the computation, the second value is the dimension of the Selmer group. The third value gives a presentation of the (fake) Selmer group (but is only defined when UseUnits is true). It is a triple <seq1, seq2, seq3>, where seq1 is a sequence of elements in L = Q[x]/(f(x)) representing a basis of some subgroup S1 of L^*/(L^*)^2, seq2 and seq3 are sequences of integers representing bases of subgroups S2 and S3 of Q^*/(Q^*)^2, respectively, and the (fake) Selmer group S fits into the exact sequence 0 --> S3 --> S2 --> S1 --> S --> 0. This data is in terms of the possibly modified model of the curve that the algorithm actually uses. As a fourth value, a set of the number fields used in the function is returned.`(x(�(Q���/���L1\0@1H-R
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__sig__5�x.�.�����NReturn the indexed set of points on K with first three coordinates given by s.��������������(//x./�
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__sig__7�x./�����JReturn the indexed set of points on J mapping to P on Kummer surface of J.������������x30/x.$/�
__sig__8�x.,/�����JReturn the indexed set of points on J mapping to P on Kummer surface of J.�������������3@/x.4/�
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	__sig__21�	__si�x.�/��������	Igusamial�+The homogeneous defining polynomial for K.  degree ��?�����������/x.�/���
	__sig__25	\��x.�/����x.�4Extends the base field of the Kummer surface K to F..T.�	__sig__19eudoAddMultiple�"gGiven the images on the Kummer surface of points P, Q, P-Q on the Jacobian, returns the image of P+n*Q.���������������/x.�/�
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	__sig__11I\0 1����j��Given an elliptic curve over the rationals of the form y^2 = x^3 + ax + b with integers a and b, this computes the dimension of its 3-Selmer group. The first value returned is the Selmer rank (i.e., the bound for the Mordell-Weil rank one gets from the Selmer group), the second value is the GF(3)-dimension of the Selmer group itself. The parameter Bound is passed on to the class group computation. The parameter Method specifies how to deal with the global restriction involving the algebra B. If Method = 0, use class and unit groups and ideal factorisation. If Method = 1, use class and unit groups and reductions mod primes. If Method = 2, use test on cubes. �xp1/usr/local/magma/package/Geometry/CrvG2/torsion.m`F	�G	q1/usr/local/magma/package/Geometry/CrvG2/torsion.m�F	�G	rOThe rational 2-torsion subgroup of J for curves of genus 2 in simplified model.�)�)�)s�k����������1�1�180]��������d1^
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__sig__1.�1�R\R�����3�HackobjPrintSpcHypElt�=���������XR�������������[xRRlR<B�
__sig__2C�C�RtR����(F� Assign names to elliptic points.FHF��*������������c�RR|R4M�
__sig__3MHM�R�R����`M^�LtM����_HackobjPrintGrpPSL2`��������pMq
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__sig__1d�L�M�����</usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/creation.mK	 R	�4The projective special linear matrix group PSL(2,Z).T�?����������N�N�N�N��
__sig__0����N�N���������������N�:The group PGL2 of a number field, as a subgroup of PSL2(R)4����������OO�N�N�
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__sig__2(�()��NO�����*�Gamma0oup�2�Given N = [n,m,p], this returns the congruence subgroup consisting of 2 by 2 matrices with integer coefficients [a,b,c,d] with a = d = 1 mod m, b = 0 mod p, and c = 0 mod n:T:�R���������0O O�NO�=�
__sig__3>�>?��NO����[email protected]�CongruenceSubgroup�A�A���������O�>�Given N = [n,m,p], this returns the congruence subgroup consisting of 2 by 2 matrices with integer coefficients [a,b,c,d] with b = 0 mod p, and c = 0 mod n, and char(a) = 1 for char a Dirichlet character mod m�M�M�MN����������M�
rpPSL2�@returns true if and only if G is equal to Gamma^0(N) 	for some N<T��$��������N�MN��
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IsGammaUpper0����������N�@returns true if and only if G is equal to Gamma^1(N) 	for some NT|��$��������(N�MNx�
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IsGammaUpper1�#��������� N�</usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/coercion.mU	�X	�</usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/coercion.mU	�X	�
__sig__0SL21�1�The element of G defined by S.t7��$�������HN8N<N�;�
__sig__0<�<=�8NDN�����>�HackobjCoerceGrpPSL2?@���������@N�>/usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/comparison.m	�X	�>/usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/comparison.m	�X	�Returns true if A is in G.J$J�XNdN����|M�
GammaUpper0(����������O�*creates the congruence subgroup Gamma^1(N)l�������������O�N�O��
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	__sig__11�x��N�O����\ � dreturns a random element of the projective linear group G, m determines the size of the coefficients%�%�������������u�O�N�OL*�
	__sig__12 ,8,��N�O����|/Cusps�1�5returns the intersection of two congruence subgroups.�7������������x��O�N�OD<�
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��N4O����T�&yThe congruence subgroup Gamma_0(N), Gamma_1(N), Gamma(N), Gamma^1(N), or Gamma^0(N), when k = 0,1,2,3, or 4 respectively.��������������POHO�N<O\�
__sig__5����NDO�����1The full projective congruence subgroup Gamma(N).,L�����������XO�NLO��
__sig__6!�!�!��NTO����D#�*creates the congruence subgroup Gamma_0(N)�&'�����������hO�N\O�*�
__sig__7-l-�-��NdO�����0���������`O�*creates the congruence subgroup Gamma_1(N)<:T:������������O�NtO�=�
__sig__8>�>?��N|O����[email protected]�Gamma1,A���������xO�*creates the congruence subgroup Gamma^0(N)�F�F������������O�N�O�I�
__sig__9JKK��N�O����NA���������PB:returns the field over which the matrices of H are defined�
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ScalarField<TG��������QHJ	For x an element of the upperhalf plane, if x is a cusp, returns the value of x as an object of type SetCspElt; if x has an exact value in a quadratic extension, returns this value, as an object of type FldQuadElt; otherwise returns a complex value of type FldPrElt$<I�����������,Q�P Qp!J
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__sig__0>�>?S<QHQ����[email protected]�4/usr/local/magma/package/Geometry/LinearSys/linsys.m.m�[�T=/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/arithmetic.m	�m	U=/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/arithmetic.m	�m	W������������lQdQTQXQNX
__sig__0O�O�O��������P	 areturns a list of coset representatives of G in PSL2(Z); only defined for G a subgroup of PSL2(Z)
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__sig__1��.�P�P����|/?/usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/equivalence.m~	0?/usr/local/magma/package/Geometry/GrpPSL2/GrpPSL2/equivalence.m~	1"mReturns true if A is equivalent to B under action of G, and if they are, also returns a matrix g with g*A = B�"2���$��������P�P�P�P�'3
__sig__0)<)T)4�P�P�����*56Returns true if A is equivalent to B under action of G806���$��������S�P�P�P�77
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	%.�returns a sequence of points in the upperhalf plane union cusps, such that the geodesics between these points form the boundary of a fundamental domain for G&��R���������P�P�P�P'
__sig__0(�P�P�����An elliptic point.��������������d�RR�R�
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�:/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/boolean.m؁	 �	�:/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/boolean.m0�	 �	�KReturns true if and only if the element x of the upper half plane is a cuspTt��$���������R�R�RL�
__sig__0\t��R�R������KReturns true if and only if the element x of the upper half plane is a cusp,#D#��$�����������R�R�R�(�
__sig__1)�)�)��R�R����,�4Returns true if and only if z is a cuspidal element.1P1��$���������R�R�R�9�
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<�computes the intersection in the upper half plane of the two geodesics x, y, where x and y are specified by their end points, given as cusps. If the geodesics intersect along a line the empty sequence is returned.������R��������PT@T0T4T�#
__sig__0%�%�%0T<T�����(GeodesicsIntersection�)��������8T8�computes the intersection in the upper half plane of the two geodesics x, y, where x and y are specified by their end points. If the geodesics intersect along a line the empty sequence is returned.�;�����R��������XT0TLT?
__sig__1?�?@0TTT����pA9/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/metric.mؔ	��	9/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/metric.m0�	��	����������pT`TdT�K
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;/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/creation.m(�	�	+AngletHyp�$rThe set of points in the complex plane which are rational or have positive imaginary part, together with infinity.L?d?����������XSHSLSC�
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__sig__3T|�HS�S������HackobjCoerceSpcHyp4����������S�3An nth root of unit in H, where n is in 3, 4, or 6.�#�#�������������SHS�S)�
__sig__4)4*L*�HS�S����D-�>/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/equivalence.m	�	�>/usr/local/magma/package/Geometry/GrpPSL2/SpcHyp/equivalence.m	�	�8�for a congruence subgroup G, finds whether the cusps a and b are equivalent under the action of G, and if so returns true and a matrix in G taking a to b. If not, returns false and the identity.|@�@����$��������S�S�S�S,E�
__sig__0FlF�F��S�S�����G�8�for a congruence subgroup G, finds whether the cusps a and b are equivalent under the action of G, and if so returns true and a matrix in G taking a to b. If not, returns false and the identity.�PQ����������pSHSdS�HSlS������$�������T�S�S�S��
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��S�S�����B�returns a point z equivalent to x under the action of PSL(2,Z), which is in the standard fundamental domain given by the region -1/2 < x <= 1/2 and |x| >= 1 for x >= 0, and |x| > 1 for x < 0, and also the matrix g in PSL(2,Z) with g*x = z\�����������S�S�S��
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__sig__34\5�5�ST�����7F�For a congruence subgroup G and edges a and b, which are given by pairs of cusps, return true or false depending on whether the edges are equivalent under the action of G, and if they are, also return a matrix g with g*a=b (g is not necessarily unique)A�A�����$������� TT�STF
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__sig__8w$U�U����x4�Returns a sequence of pairs of cusps which are cusps of the Farey Symbol FS, and which are not adjacent in FS but which are images of 0 and infinity under some matrix in PSL_2(Z).y�R���������U$U�U{$U�U����|

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	__sig__10�$UV�����\returns the index in PSL2(Z) of the congruence subgroup corresponding to the Farey Symbol FS�����������V$UV�
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	__sig__22rus8,i9The degree of M, which is the dimension of the root of M.��	x�	h����������\��}�z�}Ѕ	j�z�}�����
rceModSymlDimensionComplexTorust>m���������}o$vTrue if and only if M is the ambient space of modular symbols, which was created by specifying a weight and character.p�z�}����Gs6The ambient space of modular symbols, in which M lies. and r����������h��}�z�}actet�z�}����w(�True if and only if the Hecke operators T_p, with p prime to the level of M, do not decompose M into smaller modular symbols spaces.�	Ј	v�$����������}�z�}y
	__sig__23���x�z�}����\�{UTrue if and only if M is contained in the new cuspidal subspace of the ambient space.ly if z�$���������}�}�z�}bols}
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HeckeBound�-._ ��������P�a `bCompute a matrix representing the Hecke operator T_n on the dual representation of M. This function is takes significantly less time to run than HeckeOperator(M,n) when the dimension of M is small relative to the dimension of the ambient space containing M. Note that DualHeckeOperator(M,n) is not guaranteed to equal the transpose of HeckeOperator(M,n).8CPCc __sig__8���pEd 

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__sig__9 matrix reph �������� repm z�The matrix that represents the qth Atkin-Lehner involution W_q on M, when it is defined on modular symbols. The involution W_q is defined on modular symbols when M has trivial or quadratic character and even weight (otherwise it doesn't preserve M). When possible, the Atkin-Lehner operator is normalized so that it is an involution; such normalization may not be possible when the weight k of M is >2 and the characteristic of the base field of M divides q.tgj DualHeckeOperatoripik ����������l �����������������o 
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__sig__5��� RestrictionOfScalarsToQ� ��������d ������������������L|�s, w# V9Let H be a cyclic subgroup of order ord of (Z/NZ)^*. This function computes the space of modular symbols corresponding to the subgroup Gamma_H of Gamma_0(N) of matrices that that modulo N have lower-left entry in H. This space corresponds to the direct sum of spaces with Dirichlet character that is trivial on H.sig__2% 
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	__sig__54��z����� ��������������z���� �z�z����\��4�The direct sum of the spaces ModularSymbols(eps,k,sign), where eps runs through representatives of the Galois orbits of the characters in chars. This is a spaced defined over Q.��DL�������	
 
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	__sig__42��	��	 �z������ bRestriction of scalars down to Q. Here M must be defined over a finite extension of the rationals.runs thro����������� �L`.	 
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estriction NRestriction of scalars to Q of the direct sum of the given modular symbols spaces, which are assumed distinct. The modular symbols spaces must be defined over a finite extension of the rationals, and must not be multi-character spaces. The level, weight, and sign must be constant.	\� L4�����X� HeckeOperatorModSym��� ��������8�� 1/usr/local/magma/package/Geometry/ModSym/period.m��	�	� 1/home/was/magma/packages/ModSym/code/qexpansion.m@�	�	� .�A surjective linear map from the ambient space of M to a vector space, such that the kernel of this map is the same as the kernel of the period mapping.Note that M� 
__sig__1ional numbe� P�|����� of � 0�The dual modular symbol associated to M, viewed as a map of Hecke modules M_k ----> M_k/Ker(Phi_M), where the quotient is viewed as an abstract *vector space.*l�p��� 
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__sig__5�	�[�I 4�The images of the ith standard basis vector under the Hecke operators Tp for p<=n prime. These are computed using sparse methods that don't require computing the full Hecke operator.H ���R��������(���$�l\� P������phi_�  1/home/was/magma/packages/ModSym/code/qexpansion.m [email protected]�	`�	� ��������������l���� �������������P� ModularSymbolOddXp� ��������Ђ� <�The q-expansion of one of the Galois-conjugate newforms associated to M, computed to absolute precision prec. The coefficients of the q-expansion lie in a quotient of a polynomial extension of the base field of M.l�� 
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__sig__5� ��d�����L�L IntegralHeckeOperator�bM ���������O 0�Computes the Hecke module over the base *field* generated by a vector v. The result is returned as a subspace of the vector space underlying the ambient space.���P ��D������[�� 
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__sig__2�	�k	� bkMultiplication by X^pY - XY^p, which is a possible analogue of the theta-operator. (On mod p modular forms, the theta-operator is the map given by f |--> q df/dq.) Both M_1 and M_2 must be spaces of modular symbols over a field of positive characteristic p; they must have the same level and character, and the weight of M_2 must equal the weight of M_1 plus p+1.possible� �����������@���<�0�	C ZGA surjective linear map from the ambient space of M to a vector space, such that the kernel of this map is the same as the kernel of the period mapping. Note that M must be defined over the rational numbers. This map is normalized so that the image of IntegralBasis(CuspidalSubspace(AmbientSpace(M))) is the standard Z-lattice.�؁p&� 
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__sig__1[�__si� �����������(���$�G� ��,�����<H0!__sig__0����3!����������4!3/usr/local/magma/package/Geometry/ModSym/subspace.mH	�	k!./home/was/magma/packages/ModSym/code/verbose.m	�	0	�[�6!��������������܄8!�������Ԅ;!WThe Eisenstein subspace of M. The is the complement in M of the cuspidal subspace of M.AThe :!���������������=!
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qIntegralBasis� ��������H�� �R����������x���t� of � :�The integral basis of q-expansions in reduced form for the space of modular forms associated to M, computed to absolute precision prec. The base field must be either the rationals or a cyclotomic field.� ��������&	� @The q-expansion of the newform attached to the elliptic curve E.&	k	� �N����������������� ��������%� \Exactly the same as CompactSystemOfEigenvalues, but returns only the vector and not the map.ons in � ��R��������ȃ��ăf M.� ��̃����is p� r�Elements [v_2, v_3, v_5, v_7, ...,v_p] of a vector space and a map psi such that psi(v_i) = a_i, where a_i is the ith Fourier coefficient of one of the newforms corresponding to M. The prime p is the largest prime <= prec. This intrinsic takes far less memory and the output is MUCH more compact than SystemOfEigenvalues. To get everything over Q when M is defined over a cyclotomic extension, use CompactSystemOfEigenvaluesOverQ.d. Th�DNThe system of Hecke eigenvalues [a2, a3, a5, a7, ..., a_p] attached to M, where p is the largest prime less or equal to prec. The a_i lie in a quotient of a polynomial extension of the base field of M. It is assumed that M corresponds to a single Galois-conjugacy class of newforms.�!��������������|�'!
__sig__6���&!�������^Y!V>The p-new subspace of M. This is the complement in M of the subspace generated by the modular symbols of level equal to the level of M divided by p and character the restriction of the character of M. If the character of M does not restrict, then NewSubspace(M,p) is equal to M. Note that M is required to be cuspidal._!0�The new subspace of M. This is the intersection of NewSubspace(M,p) as p varies over all prime divisors of the level of M. Note that M is required to be cuspidal.orm attac^!��������������|�a!
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IsLinearScheme� �!��������܆�!2True iff the ambient space of X is two dimensionalh'�'�!|$����������D����+�!
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schemeZ!��l�����0"]The reduced subscheme of X defined by its reduced scheme structure, followed by the map to X.�	"||x�������̇��"̇������"ReducedSubscheme��"���������"
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__sig__0nents[email protected]"CA sequence containing the minimal prime components of the scheme X.,DDD"|R��������l�̇`��G"
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0/usr/local/magma/package/Geometry/Sch/jacobian.mX�@	�D	 "The inflection points of Cd[�[!"s|����������|���Xc#"|��������e$"InflectionPointsg�g�g%"����������&"The inflection points of Cnn'"s|����������|���,q("
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	__sig__14P0�0�"4�P������5�"__sig__1Ove��	�	�m	�[��"&}A sequence of the singular points of X defined over the base ring of X, assuming that the singular locus is zero dimensional.>�"|���������d�4�X��@�"4�`�����C�" SingularPointsOverSplittingFieldE�E�E�"��������\��"5/usr/local/magma/package/Geometry/Sch/constructions.m	�	�"
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__sig__1�Ԋ��#D�h�������#SetAFR���#"hTrue iff there is an element of the function field C corresponding to f together with that element if so
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�#��s$����������x�#���������#��������d��#1/usr/local/magma/package/Geometry/SrfK3/baskets.m��	��	�#1/usr/local/magma/package/Geometry/SrfK3/baskets.mH�	��	�#z�The first return sequence contains baskets of singularities for virtual K3 surfaces polarised by a divisor with a g-dimensional linear system (or bigger) and at most n curves in the resolution of its singularities where g is the corresponding entry of the second sequence. The third sequence is the corresponding degree, while (if Proof is true) the fourth contains elements which had positive degree but negative coefficients for the indicated value of g����#�RRRR����|���D#�#
__sig__0$%0%�#|��������'�#�����������#1/usr/local/magma/package/Geometry/SrfK3/centres.m��	��	�#1/usr/local/magma/package/Geometry/SrfK3/centres.m��	��	�#RDestructively compute centres of X that have a plausible target in the sequence DB;$;�#R=�������������������>�#
__sig__0?�?�?�#���������@�#Centres\B�#�����������#ZDestructively compute centres of K3s in the sequence DB that have a plausible target in DBTHlH�#R����������,�����M�#
__sig__1N�N�N�#��������Q@$codimies�# `Compute a sequence containing the numbers of K3 surfaces in DB which might be unprojections of XY�Y�Y�#R=�����������ԏ��ȏ�^�#
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Unprojections�g�#��������̏�#=Destructively compute unprojections of K3s in the sequence DB,o�#R����������|�������r�#
__sig__3st0t�#�������@v�#<A coded sequence of chains of unprojections within DB from X}H~�#R=R�����������������#
__sig__4��p��#��������X�$UnprojectionChains���$���������$0A sequence of coded chains of projections from X������(��#
IsCurveFFElt<�#����������#$oAn element of the function field of the ambient of C corresponding to the quotient f of two ambient polynomials�<�#��|���������$����4�#
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AmbientFFElt#D#�#����������#8�An element of the function field of the ambient of C corresponding to f (which can be a curve function field element, an ambient function field element or a quotient of two ambient polynomials)�1�1�#��s$������<���0�$:�#
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CurveFFElt\>t>�#��������4��#$qTrue iff f is in the coordinate ring of the ambient space of S, in which case return the corresponding polynomial�E�E�#��|$?������T���H�lH�#
__sig__8I�I�I�#��P����� L�#IsAmbientFunctionlN�N�#��������L��#
__sig__0Function�#=True iff f in is the function field (or coordinate ring) of AY�#��|$��������l���`�(^�#
__sig__9_�_�a�#��h������d*$HGiven a sequence of polynomials determining a Hilbert polynomial with periodic corrections, together with a sequence of alternative early values, return the Hilbert series' numerator, together with a sequence containing the factors of the minimal denominator$+$RR8R�������������-$��������.$+HilbertSeriesMultipliedByMinimalDenominator8/$����������0$"lThe Hilbert function corresponding to a sequence of univariate polynomials and a sequence of starting values1$RR)��������̐����H#3$��Ȑ����l%4$
HilbertFunction�'5$��������Đ6$-/usr/local/magma/package/Geometry/SrfK3/two.m	��	7$-/usr/local/magma/package/Geometry/SrfK3/two.m	��	8$(�Adds additional weights to X to account for the polarisation of the singularities disregarding results of codimension bigger than cmax<9$����������ܐ��h?:$
__sig__0@[email protected][email protected];$ܐ�����`B<$ForceSingularities0DHD=$���������>$./usr/local/magma/package/Geometry/SrfK3/type.m	��	?$h./usr/local/magma/package/Geometry/SrfK3/type.m	��	B$genus�MH$steps�C$
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K3Database��O$
hilbert_coeffs0oA$���������������P$ _The database of numerical K3 surfaces; the string t will be the variable name of Hilbert series$xlx�xQ$*R��������\�L���@�ĀS$��H������V$&|The database of numerical K3 surfaces having codimension at most n; the string t will be the variable name of Hilbert Series��W$*�R��������l�d���X�d�X$
__sig__1�����#IsAmbientRationalFunction�h�h�#��������d��#./usr/local/magma/package/Geometry/Sch/compat.m	8�	�#./usr/local/magma/package/Geometry/Sch/compat.m	8�	�#-True iff X is nonsingular and equidimensional4t�#|$������������|���,z�#|��������~�#

IsNonSingularL��#�����������#
�����#y|$������������|������#
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__sig__2�����$����������$%The weights of Noether variables of X0
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	__sig__20���$���������$
NoetherForm���$����������$&The weights of Enriques variables of X��$=R��������������$
	__sig__21h��$���������$��ܒ������$)The steps in determining the weights of X�"�"�$=R�������������'�$��������)�$����������$(A sequence of possible centres of Gorenstein projections of X together with the numbers of possible images of those projections�7�7�7�$=R��������4���(�H<�$
	__sig__24x=�=�$��0�����8?�$ Remove the centres computed on X@�@A�$=����������T�D���8�pE�$
	__sig__25�F�F�$��@������G�$

RemoveCentres8I�$��������<��$4Remove the centres computed on the K3 surfaces in DBO�O�$R����������\���P��U�$
	__sig__26�W�W�$��X������Y�$*�A sequence of possible images among DB of Gorenstein projections of X together with the numbers of possible images of those projectionstd�d�d�$=RR��������l���`��i�$
	__sig__27�kl�$��h�����xn�$��������d��$Number0q�$4A sequence of numbers of possible unprojections of Xtlt�$=R������������x�hz�$
	__sig__28�~�~�$��������|��$The degree of V�������$=����������������<��$
	__sig__29t����$����������$%The number of X in the classification���$
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Identifier��$R=R������������0
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__sig__5(X$��������$ProjectionChainsPx$���������$ATrue iff there is a centre of Type i among the known centres of X`x	$�=$��������,��� ��
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__sig__6��$��(������$=R����������������$���������
�$�����������$%The number of X in the classification��$=R��������������l�$
	__sig__31\t�$����������$�����������$CThe index of X in the Altinok-Fletcher-Reid lists, 0 if not definedd��$����������ԓ��ȓ��$
	__sig__32�!"�$�������#�$
	AFRNumberH%h%�$��������̓�$CThe index of X in the Altinok-Fletcher-Reid lists, 0 if not defined�,D-�$=���������������5�$
	__sig__33�7�7�$�������T:�$-The index of X in the Reid codimension 1 list�<�$���������<�������?�$
	__sig__34�@�@�$��������C%

ReidNumber�D�D%�����������'H2_G_Aule%1The index of X in the Fletcher codimension 2 listJ$J%���������L�����lO%
	__sig__35�PQ%�������tT%
FletcherNumber�V%���������%GThe index of X in the Altinok codimension 3 and 4 and exceptional listsX^p^�^	%���������\�,��� ��f
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	__sig__36hHh%��(�����k%

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%��������$�%-The index of X in the Reid codimension 1 list�q%=���������D���8�v%
	__sig__37y0y%��@�����x}%1The index of X in the Fletcher codimension 2 list����%=���������T���H���%
	__sig__38 �8�%��P�������%GThe index of X in the Altinok codimension 3 and 4 and exceptional listsP�x���%
	__sig__39p���%��`�����H�$

HasCentreType�
$��������$�$*�True iff there is a centre of Type I among the known centres of X and all such centres have an image in codimension one less than that of XD(\($=$��������D���8�8,$
__sig__7.l.�.$��@������1$HasGoodTypeICentres7D7$��������<�$./usr/local/magma/package/Geometry/SrfK3/hilb.m	x		$./usr/local/magma/package/Geometry/SrfK3/hilb.m	x		$>Degree of the polarising divisor on a virtual K3 with data g,BlA$R������������d�T�X��E$
__sig__0F�FG$T�`�����<H$F�Given integer g and basket of singularities B in the form [r,a] corresponding to the polarised quotient singularity 1/r(a,-a) return the Hilbert series of the corresponding K3 surface (whether it exists or not) polarised by a divisor with g + 1 sectionsdV$R�������������t�T�h��Z$
__sig__1\�\�\$T�p������_($
__sig__0ipliedUp$ `Return a good candidate for the multiplied up Hilbert series, where g,B are as in Hilbert Seriesij4j$R���RR������T�x�,o $
__sig__2p4pLp!$T��������q"$HilbertSeriesMultipliedUpt0t#$��������|�$$-/usr/local/magma/package/Geometry/SrfK3/sum.m	x		%$-/usr/local/magma/package/Geometry/SrfK3/sum.m	x		&$ cThe Hilbert series corresponding to a sequence of polynomials F and a sequence of starting values Vp���'$RR���������l��������)$���������2$
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__sig__1���m%JDisplay the Burnside matrix corresponding to the lattice of subgroups of G��n%������������������p%��������8q%BThe Burnside matrix corresponding to the lattice of subgroups of G��r%������������Е��ĕ`t%��̕����u%
BurnsideMatrixv%��������ȕw%BThe Burnside matrix corresponding to the lattice of subgroups of G��x%��������������ܕ#y%
__sig__4$�$�$z%��������'{%?The Burnside matrix corresponding to the lattice of subgroups L�*�*�*|%%���������������1}%
__sig__55�5<6~%��������|8�+qpMatldc�+
__sig__0iliser.m%5/usr/local/magma/package/Group/Grp/is_conj_subgroup.m	�	�%5/usr/local/magma/package/Group/Grp/is_conj_subgroup.m	�	�%MWhether a conjugate of N is a subgroup of M, and if so, a conjugating elementE�%������$������������G�%������pI�%IsConjugateSubgroupK K�%����������%+The set of conjugates of H by elements of GPRhR�%����S��������(���hX�%
__sig__1Y�Y�Y�%�$����� \�%+The set of conjugates of H by elements of G�_�a�%����S��������8��,��g�%
__sig__2hipi�%�4�����l�%//usr/local/magma/package/Group/Grp/commutator.m�	�%//usr/local/magma/package/Group/Grp/commutator.m�	�%5The commutator group [H, K] = <(h,k): h in H, k in K>lt�%��������������P�@�D�hz�%
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CommutatorGroup���%��������H��%./usr/local/magma/package/Group/Grp/smallgps2.m	�	�%Z./usr/local/magma/package/Group/Grp/smallgps2.m	�	�%
__sig__4abase��%HReturn true iff the small groups database contains the groups of order o�4�L��%__sig__0����$�<��$EThe element of the K3 database DB with codimension c and AFR number n0
�$��R=��������4���(�8
�$��0�������$K3SurfaceFromAFR �$��������,��$3The element of the K3 database DB with identifier i��$�R=��������L���@��$
	__sig__12��$��H�����x�$AA record of the attributes of V in the format of a database entryH!p!�$=��������\���P�l%�$
	__sig__13�'�'�$��X������)�$
NumericalRecord�*�$��������T��$The codimension of X1�1�$=R��������t���h�(:�$
	__sig__14H;`;�$��p������<�$<The number g associated to the polarising linear system on X?�?�$=���������������x�PC�$
	__sig__15E0E�$���������F�$ cThe Hilbert series of the scheme X after cutting by hyperplane sections of degrees the weights of X@JXJ�$=8���������������O�$
	__sig__16Q4Q�$���������T�$HilbertNumeratorWxW�W�$�����������$QThe weights of variables of X before unprojection conditions have been considered�_�a�$=R���������������g�$
	__sig__17ipi�$��������l�$
HilbertFormnn�$�����������$
	__sig__23ned0q�$NThe weights of Noether variables of X if computed, otherwise an empty sequencedu�$=R��������Ē����L~�$
	__sig__18��Ā�$����������$NoetherFormIfAssigned���$�����������$*The apparent degrees of the equations of X� 	�!	�$=R��������ؒ��Ԓ�!	�$
	__sig__22��$
EnriquesForm�L�Z$%The database of numerical K3 surfaces0[$R��������|�t���h� ]$��p������
^$BThe database of numerical K3 surfaces having codimension at most n8P_$�R������������x�`$
__sig__3(@a$���������b$Print X at level l�c$*=�����������������d$
__sig__40e$���������!f$
PrintVSrfK3�"#g$����������h$*Numerical K3 corresponding to the record X@)X)i$=����������������X.j$
__sig__50,1T1k$��������<6l$
	K3Surface�7�7m$����������n$2Numerical K3 with genus g and singularity basket Bx=�=o$R�=��������D����@p$
__sig__6A�A`Bq$��������HD�$
	__sig__10et�Er$1The elements of the K3 database DB with weights W�GHs$RRR��������ԑ��ȑ$Lt$
__sig__7N0NPNu$��Б����8Pv$K3SurfacesFromWeights�Sw$��������̑x$/The unique K3 in the database DB with weights W�YZPZy$RR=��������������az$
__sig__8dTdtd{$�������g|$K3SurfaceFromWeightsh�h}$���������~$<The elements of the K3 database DB with singularity basket Bo�o$RRR�������������ls�$
__sig__9t�t�t�$��������x�$K3SurfacesFromBasketzl|�$�����������$:The unique K3 in the database DB with singularity basket B����$RR=��������������$�������D��$K3SurfaceFromBasket�4��$����������$
	__sig__11��܏�$Images��(��@�����(
	AddPrimes (��������<�(Emp is a set of primes which are added to the relevant primes for SQP.p(S�����������\���P��(
__sig__6�(��X�������9IsEqualct(;Replace the set of relevant primes in SQP by the given set.(S�����������l���`�@(
__sig__7�(��h������(

ReplacePrimes�(��������d�(.Check the correctness of the soluble quotient.�.(�$������������x��= (
__sig__8=�=!(���������="(
SQ_check@�@#(��������|�$(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.$M%(��$�������������XM&(
__sig__9RR'(��������4R((EquivalentQuotients<U)(����������*(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.0a+(���������������da,(
	__sig__10xa-(���������a.(IntersectKernelsa�a/(����������0(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.�z1(���$�������̠����{2(
	__sig__11{3(��Ƞ����,{4(ComposeQuotients{D{5(��������ĠN(
	__sig__17cessx{6(&xStart the soluble quotient algorithm for a finitely presented group F without any information about the relevant primes.{�{7(y�������������ؠ�{8(
	__sig__12|9(�������$|:(SolubleQuotientProcess<|;(��������ܠ=(�y����������������p(Modulesr�%NumberOfSmallGroups���%��������`��%FThe number of groups of order n stored in the database of small groups��%������������`�t���%
	__sig__15���%`�|�������%
	__sig__18Limit�%EThe order up to which the small groups database D contains all groups��%5�������������`�����%
	__sig__16���%`���������%SmallGroupDatabaseLimit"�%�����������%CThe order up to which the small groups database contains all groups))�%�����������`���.�%
	__sig__17T0�0�%`��������5�%?Return the small groups database, opened for an extended searchX:p:�:�%5����������`���>�%`�������h?�%SmallGroupDatabase�@�@�%����������&
	__sig__22roup0E�%?Return the small groups database, opened for an extended search�G�G�G�%5��������З`�ė�K�%
	__sig__19NN�%`�̗�����O�%OpenSmallGroupDatabase`S�%��������ȗ�%Close the smallgroup databasePY�%5�����������`�ܗ\^�%
	__sig__20�a�b�%`������Le�%CloseSmallGroupDatabase�g�%�����������%7Moves the small group process tuple p to its next group�n�n�n�%/�����������`���0r�%
	__sig__21�s�s&`�������v&InternalNextSmallGroup|y&����������&	__sig__23ProcessRestart�&:Returns the small group process tuple p to its first group����&/�����������`����&`�������& InternalSmallGroupProcessRestart�|���&���������
&�����������@�0�`�$�L��'Rootcts%=���������d���X�X%The Hilbert series of X��
%=���������|�t���h� %
	__sig__40�%��p�����H%TThe Hilbert series of the scheme X after cutting by hyperplane sections of degrees W�%R=�������������x�� %
	__sig__41��!%��������0"%$sThe sequence of the first n coefficients of the Hilbert series of X including the constant coefficient as the firstL%l%#%�=R���������������)$%
	__sig__42�+,%%���������.&%HilbertCoefficients�1�1'%����������(% _Add at most two weights to try to make all local basket polarisations arise from global weights�<�<�<)%==��������������@*%
	__sig__43A0A+%��������PC	&x�Returns a small group process. This will iterate through all groups with order in Orders. To extract the current group from a process, use ExtractGroup(). To move to the next group in a process, use NextGroup(). To find out which group the process currently points to, use ExtractLabel(). The user may limit the process to soluble or insoluble groups by setting Search. Search may take the values "All", "Soluble" or "Insoluble" (or variants thereof).$<&`�,������
&SmallGroupProcess��&��������(�&*�Returns a small group process as described above. This will iterate through all groups (g) with order in Orders which satisfy Predicate(g).�"#&�������������P�H�`�<�\(&
	__sig__24l)�)&`�D������+& dReturns a small group process as described above. This will iterate through all groups with order o.5�5&����������`�X�`�L��:&
	__sig__25�;<&`�T������=&(�Returns a small group process as described above. This will iterate through all groups (g) of order o which satisfy Predicate(g).�BC&������������h�`�\��F&
	__sig__26�G�G&`�d�����lI/&	__sig__30llGroupLabelyK&HReturns true if the small group process tuple has passed its last group.OP4P&/$��������x�`�l��V&
	__sig__27|X�X&`�t�����LZ& InternalSmallGroupProcessIsEmpty\�\(^ &��������p�!&<Returns the current group of the small group process tuple pg�g"&/������������`����m#&
	__sig__28�n�n$&`�������Lp%&InternalExtractSmallGroupr,r&&����������'&$rReturns the label (s,n) of the small group process tuple p. This is the order and number of the current group of p�~�(&/����������`���h�)&
	__sig__29@�X�*&`�������8�+&InternalExtractSmallGroupLabel`�,&����������.&���5����������Ș��`���؏0&`�������ؑ3&
	__sig__31����4&`�̘����ؔ,%AddLocalGenerators,GDG-%����������.%ISet the identifiers of the elements of K3 to be 1..#K3 in the given order�MN/%R����������Ĕ�����S0%
	__sig__44xU�U1%��������dX?%
	__sig__51grees�Y2% aThe apparent degrees of equations from the initial negative coefficients of the Hilbert numerator�a�b3%R���������Ԕ��Ȕ�g4%
	__sig__46�i�i5%��Д����Tl6%ApparentEquationDegrees�n7%��������̔8% aThe apparent degrees of equations from the initial negative coefficients of the Hilbert numeratorHu`u9%=R�������������H~:%
	__sig__47����;%��������=% _The apparent codimension according to the sign changes of coefficients of the Hilbert numerator� �8�<%=R����������������>%����������@%ApparentCodimensionȎ��A%���������E%
	__sig__52��ȒD%���������-&B�Returns the first group 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).Lt1&B�Returns the first group 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).��2&�������������ؘИ`�Ę�5&(�Returns the first group with order in Orders which satisfies the predicate and the search condition specified by Search (see above)."|"6&��R�������������`�Ԙ�'7&
	__sig__32�()8&`�ܘ�����*9&(�Returns the first group with order in Orders which satisfies the predicate and the search condition specified by Search (see above).6�6:&��R5�������������`��\;;&
	__sig__33|<�<<&`������t>=&"hReturns the first group of order o which satisfies the search condition specified by Search (see above).BtB�B>&�������������`����F?&
	__sig__34�G�G@&`�������4IA&"hReturns the first group of order o which satisfies the search condition specified by Search (see above).N$OlOB&�5�����������`��PUC&
	__sig__35tW�WD&`�������YE&`cReturns a list of all groups of order o. The user may limit the search to soluble or insoluble groups by setting the parameter Search. Search may take the value "All", "Soluble" or "Insoluble" (or variants thereof). Some orders will produce a very large list of groups -- in such cases a warning will be printed unless the user specifies Warning := false.�n�nF&�5���������0� �`��,rG&
	__sig__36�s�sH&`������vI&
SmallGroupsy0yJ&���������K&`cReturns a list of all groups of order o. The user may limit the search to soluble or insoluble groups by setting the parameter Search. Search may take the value "All", "Soluble" or "Insoluble" (or variants thereof). Some orders will produce a very large list of groups -- in such cases a warning will be printed unless the user specifies Warning := false.�(�L&����������@�8�`�,��M&
	__sig__37 �H�N&`�4�����(��&
phismsg�%�$����������p�`�d�l
�%`�l�������%IsInSmallGroupDatabase4
�%��������h��%0Return true iff D contains the groups of order o$<�%�5$����������`�|�<�%
__sig__1Tt�%`���������%DReturns the group number n of order o from the small groups database�%����������������`���`��%.Return true iff SmallGroup(D, o, n) is solubleh%�%��5$����������`����)�%
__sig__3+�+,�%`��������.�%`��������1�%
__sig__6ble6�6�%GReturn true if SmallGroup(D, o, n) is soluble (does not load the group)$;D;\;�%��5$������������`����>�%
__sig__5?�?�?�%`�������,A�%SmallGroupIsSolvableCLC�%�����������%GReturn true if SmallGroup(D, o, n) is soluble (does not load the group)lH�H�H�%��5$���������Ж`�Ė�M�%`�̖�����O�%SmallGroupIsSolubleLRdR�%��������Ȗ�%
__sig__8bleXdX�%DReturn true if SmallGroup(o, n) is soluble (does not load the group)\l\�%��$���������`�ܖpd�%
__sig__7eHf�f�%`�������h�%DReturn true if SmallGroup(o, n) is soluble (does not load the group)m�m�%��$����������`���p�%`��������r�%
	__sig__13bleht�%FReturn true if SmallGroup(o, n) is insoluble (does not load the group)(z�%��$��������0��`������%
__sig__9������%`������(�{&LReturns a sequence of names for the simple groups that have order dividing n�|&�R����������Й��~&Й������8&SimpleGroupsWithOrderDividing��&����������&2�The order of the simple group specified by the tuple T. T is assumed to be a tuple as returned by IsSimpleOrder or as part of the sequence returned by CompositionFactors��&/����������Й���&Й�������&SimpleGroupOrder!�!"�&����������&
__sig__0ble%p&�&5Can the Subgroups family of functions be applied to G�)�&��$��������(���/�&
__sig__31�1�1�&Й$�����H7�&IsSubgroupsAvailable9�9�&�������� ��&1/usr/local/magma/package/Group/GrpAb/direct_sum.m�}	��	�&1/usr/local/magma/package/Group/GrpAb/direct_sum.m�}	��	�&&zThe direct sum of the abelian groups in S followed by the sequences of canonical inclusions and projections, respectively.�F�F�&�kkRR����t�H�8�<��I�&8�D������K�&*/usr/local/magma/package/Group/GrpAb/hom.m`	��	�&*/usr/local/magma/package/Group/GrpAb/hom.m�	��	�&:�An abelian group A isomorphic to Hom(G, H) (where G and H are finite abelian groups), together with the transfer map t which, given an element of A, returns the corresponding group homomorphism from G to H�^�&kkk�������h�`�P�T�g�&
__sig__0h�h�h�&P�\�����xk�&<�An abelian group A isomorphic to Hom(G, H) (where G and H are finite abelian PC groups), together with the transfer map t which, given an element of A, returns the corresponding group homomorphism from G to HtTtlt�&��k�������T�p�P�d�hz�&
__sig__1~�~�~�&P�l�����|��&(�A sequence of (Z-module) generators of the set of all homomorphims from the finite abelian group G to the finite abelian group H�T����&kkR��������$���P�t���&
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HomGeneratorsL��&��������x��&Ecentral extension of (abelian) U by G determined by cocyclic matrix AP��%SmallGroupIsInsolvable���%����������%FReturn true if SmallGroup(o, n) is insoluble (does not load the group)��%��$��������@� �`��̒�%
	__sig__10���%`��������&�����������������te a�&
__sig__3te abelian �&�������.CentralExtension����&\A sequence of all homomorphims from the finite abelian group G to the finite abelian group H�&kkR������������P�����&
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Homomorphisms��&�����������& bA sequence of all homomorphims from the finite abelian PC group G to the finite abelian PC group HT"|"�&��R����������P����'�&
__sig__5(�()�&P��������*�&OBSOLETE: Use Homomorphisms�-.�&kkR����������КP�Ě�6�&
__sig__67X8x8�&P�̚�����:�&AllHomomorphisms<,<D<�&��������Ț�&OBSOLETE: Use Homomorphisms�?@�&��R���������P�ܚD�&
__sig__7E�E�E�&P������G�&-/usr/local/magma/package/Group/GrpAb/subgrp.m	�	�&-/usr/local/magma/package/Group/GrpAb/subgrp.m	�	�&
__sig__1nGroupLN�&k���������������S�&
__sig__0U<VdV�&��������X�&SubgroupsOfAbelianGroupLZ�&�����������&1Compute the subgroups of the finite abelian groupPdpd�&kR�����������li�&������l�&8/usr/local/magma/package/Group/GrpData/perfgps/perfgps.mo�	�	�&8/usr/local/magma/package/Group/GrpData/perfgps/perfgps.mrp�	�	�&DReturn the i-th perfect group in the database as a permutation groupv w�&�5���������8�0� �$���&
__sig__0�`�x��& �,��������&OReturn the i-th perfect group in the database of order o as a permutation group8�P����&��5���������H�@� �4����&
__sig__1�x����& �<��������&KReturn the perfect group in the database named 'top' as a permutation group��Ȓ�&*5���������X�P� �D�Ԗ�%SmallGroupIsInsoluble�%����������%IReturn true if SmallGroup(D, o, n) is insoluble (does not load the group)�%��5$��������8�`�,��%
	__sig__11�%`�4������%IReturn true if SmallGroup(D, o, n) is insoluble (does not load the group)�%��5$��������H�`�<��%
	__sig__12�%`�D������%MThe size and number of the group isomorphic to G in the small groups database�%��/��������X�`�L��%`�T������%8The number of groups of order n stored in the database D�%�5���������x�h�`�\��%
	__sig__14�%`�d������&
__sig__2(�& �L�����,�&\Return the perfect group in the database which is 'top'#prime<exp, n> as a permutation group��&���*5���������h�`� �T���&
__sig__3���& �\�������&.�Return the permutation group image of G under the homomorphism denoted by R (which should be an element of the sequence returned by Group(D[pfgps], ...)<�&/y���������p� �d�4�&
__sig__4T��& �l�����"'
__sig__0ationsn�&&|An isomorphism from the i-th perfect group G in the database as a fp group onto a representation of G as a permutation group)�)�&�5�y��������� �t�80�&
__sig__51|2�2�& �|�����t7�&*�An isomorphism from the i-th perfect group G of order o in the database as a fp group onto a representation of G as a permutation group�=�=>�&��5�y��������� ����@�&
__sig__6BtB�B�& ��������D�&$sAn isomorphism from the perfect group G named 'top' as a fp group onto a representation of G as a permutation group�H4I�&*5�y��������� ���N�&
__sig__7O�O�O�& �������dR�&(�An isomorphism from the perfect group G which is 'top'#prime<exp, n> as a fp group onto a representation of G as a permutation groupZLZ�&���*5�y��������� ����a�&
__sig__8dPdpd�& �������g�&*�A homomorphism from G onto a permutation group P defined by R (which should be an element of the sequence returned by Group(D[pfgps], ...)�n�n�&/y���������� ���,r�&
__sig__9s�s�s�& �������v�&"hThe number of available representations of the i-th perfect group in the database as a permutation group������&�5�����������Л �ěp��&
	__sig__10����& �̛�������&NumberOfRepresentations��&��������ț�&$sThe number of available representations of the i-th perfect group of order o in the database as a permutation group��ؑ�&��5����������� �ܛX��&
	__sig__11�$��& ������$�O&$As above, but with a list of orders.,P&R5���������P�H�`�<�Q&
	__sig__38
4
R&`�D������S&$As above, but with a list of orders.|T&R���������`�X�`�L�,U&
	__sig__39T|V&`�T������W&&}Returns a list of all groups (g) with order o which satisfy Predicate(g) eq true and the search condition specified by Search4X&���5���������p�h�`�\�|"Y&
	__sig__40�#�#Z&`�d�����l&[&&}Returns a list of all groups (g) with order o which satisfy Predicate(g) eq true and the search condition specified by Search�-\&��������������x�`�l�86]&
	__sig__41�7�7^&`�t������:_&$As above, but with a list of orders.<�<`&��R5�������������`�|��?a&
	__sig__42�@�@b&`�������Cc&$As above, but with a list of orders.E�Ed&��R�����������`���lHe&
	__sig__43�I�If&`������� Lg&BEncode a pc-group as an integer using the small groups data codingLPtPh&�����������`���4Wi&
	__sig__44�XYj&`��������Zk&SmallGroupEncoding�\�\l&����������m&7Decode a small groups data code representing a pc-grouppg�g�gn&�������������`���Hmo&
	__sig__45�n�np&`�������pq&SmallGroupDecodingdq�qr&����������s&1/usr/local/magma/package/Group/Grp/simple_names.m��	Ȳ	t&1/usr/local/magma/package/Group/Grp/simple_names.m�	Ȳ	u&6�A name (and a possible alias) for the simple group specified by the tuple T. T is assumed to be a tuple as returned by IsSimpleOrder or as part of the sequence returned by CompositionFactors�v&/*����������Йԙ`�w&
__sig__0�����x&Йܙ�������&$oThe number of available representations of the perfect group named 'top' in the database as a permutation groupp
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	__sig__12$�& �������P�&"gThe number of available representations of the perfect group 'top'#prime<exp, n> as a permutation group��'���*5���������� ���'
	__sig__13�' ������x'//usr/local/magma/package/Group/GrpData/prmgps.m�	' //usr/local/magma/package/Group/GrpData/prmgps.m�	1'
__sig__8imitiveGrou'GReturns true if the primitive group process has passed its last group. �)�)�)'/$�������� ���<0'�������4	'$InternalPrimitiveGroupProcessIsEmpty8|8
'���������'<Moves the primitive group process tuple p to its next group >x>'/����������8��,�0A
'
__sig__1B�B C'�4�����0E'InternalNextPrimitiveGroup�F�F'��������0�' _Returns a primitive group process. This will iterate through all groups with degree in Degrees.N0NPN'�����������`�P��D��S'
__sig__2U@VhV'�L������X'PrimitiveGroupProcessPZ'��������H�'&~Returns a primitive group process which will iterate through all groups (g) with degree in Degrees which satisfy Predicate(g).�g'�������������p�h��\��m'
__sig__3n�n�n'�d�����Pp'TReturns a primitive group process. This will iterate through all groups of degree d.tlt'������������x��l�hz'
__sig__4~�~�~'�t�����|�'$oReturns a primitive group process. This will iterate through all groups of degree d which satisfy Predicate(g).����� '���������������|�4�!'
__sig__5�܌�"'�������\�#'AReturns the current group of the primitive group process tuple p ���$'/�*�������������%'
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SimpleGroupName��z&��������ؙ�&__sig__2rderDividingؖ}&
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__sig__2���'��̞������'��������Ȟ�'-/usr/local/magma/package/Group/GrpExt/coho2.m	0�	�'./usr/local/magma/package/Group/GrpExt/compab.m	0�	�'1/usr/local/magma/package/Group/GrpExt/compabnew.m	(�	0�	�'./usr/local/magma/package/Group/GrpExt/degred.m	0�	�'//usr/local/magma/package/Group/GrpExt/elabext.m0�	�'//usr/local/magma/package/Group/GrpExt/elabext.m0�	�'__sig__2ntaryAbelianGroup�'[All extensions (up to isomorphism) of an elementary abelian group of order p^d by a group G�/<0�'���R��������������7�'
__sig__0:@:X:�'��������;�'"ExtensionsOfElementaryAbelianGroup�=>�'�����������'"hAll extensions (up to isomorphism with N fixed) of an elementary abelian group of order p^d by a group GEE0E�'/���R�����������H�'
__sig__1IPIpI�'������ K�',All extensions (up to isomorphism) of M by GN�N�'gy��R��������(����T�'�$������W�'DistinctExtensionshY�Y�'�������� ��'./usr/local/magma/package/Group/GrpExt/modext.m	0�	�'./usr/local/magma/package/Group/GrpExt/solext.m	0�	�'./usr/local/magma/package/Group/GrpExt/solext.m	0�	�'

__sig__116Groupl�'@All extensions (up to isomorphism) of soluble group H by group Go�o�o�'��R��������L�<�@�ls�'
__sig__0t�t�t�'<�H������x�'ExtensionsOfSolubleGroup|d}|}�'��������D��',/usr/local/magma/package/Group/GrpFP/SQ/sq.m�	0�	�'

__sig__120magma/pac+Seedtsis	�'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. calculates t�'Sy��R*����|���p��'��x�������:*4/usr/local/magma/package/Group/GrpMat/Smash/larger.m�	0�	L*The block system of G��i'
__sig__3mptyPU'HReturns true if the transitive group process has passed its last group. 
$TV'/$��������`�P�T��X'P�\������Y'%InternalTransitiveGroupProcessIsEmptylZ'��������X�u'
__sig__6ocessroup['=Moves the transitive group process tuple p to its next group �\'/����������x�P�l��]'
__sig__1,^'P�t������!_'InternalNextTransitiveGroupd#�#`'��������p�a' `Returns a transitive group process. This will iterate through all groups with degree in Degrees., ,8,b'���������������P����4c'
__sig__27\7t7d'P�������$:e'TransitiveGroupProcess�;f'����������g'(Returns a transitive group process which will iterate through all groups (g) with degree in Degrees which satisfy Predicate(g).�B�BCh'�����������������P����Fj'P�������Hk'UReturns a transitive group process. This will iterate through all groups of degree d.|Ml'��������������P���0Qm'
__sig__4SDTtTn'P�������4Wo'$pReturns a transitive group process. This will iterate through all groups of degree d which satisfy Predicate(g).\�\(^p'������������ȝP����eq'
__sig__5g�g�gr'P�ĝ�����i�'__sig__8ansitiveGroupLabels'BReturns the current group of the transitive group process tuple p �opt'/�*������؝P�̝�sv'P�ԝ����vw'InternalExtractTransitiveGroup�yx'��������Нy'&yReturns the label (s,n) of the transitive group process tuple p. This is the degree and number of the current group of p p���z'/���������P���{'
__sig__7�X���|'P��������}'#InternalExtractTransitiveGroupLabelp���~'���������'&yReturns the first group (g) of degree d which satisfies Predicate, along with a string giving a description of the group.��ԗ�'P���������'__sig__9�<�T�����''InternalExtractPrimitiveGroupl('����������)'&xReturns the label (s,n) of the primitive group process tuple p. This is the degree and number of the current group of p ��*'/�������������D+'
__sig__7D\,'��������-'"InternalExtractPrimitiveGroupLabel��.'����������/'BReturns the first group (g) of degree d which satisfies Predicate.T�0'����*������МȜ����#2'�Ĝ����l&3'QReturns the first group (g) with degree in Degrees which satisfies the predicate.�*�*4'�����*��������؜�̜�15'
__sig__95�5866'�Ԝ����x87'(Returns the first group (g) of degree d.;t;�;8'��*��������ܜ?9'
	__sig__10�?@:'������lA;':�Returns a sequence of all primitive groups of degree d. Some degrees will produce a very large sequence of groups -- in such cases a warning will be printed unless the user specifies Warning := false.I�I�I<'�R��������������N='
	__sig__11P4P>'��������S?'
PrimitiveGroups�U@'���������A'%As above, but with a list of degrees.�[B'��R�������������cC'
	__sig__12`e�eD'�������gE'RReturns a list of all groups (g) with degree d which satisfy Predicate(g) eq true.�m�mF'���R��������(� ����pG'
	__sig__13Dr�rH'������htI'%As above, but with a list of degrees.�xJ'����R��������0��$���K'
	__sig__14���L'�,�����h�W'
__sig__0criptionM'FReturn the string description of the n-th primitive group of degree d.��N'��*��������@��4���O'
	__sig__15p���P'�<�����H�Q'PrimitiveGroupDescription`���R'��������8�S'//usr/local/magma/package/Group/GrpData/trngps.m��	T' //usr/local/magma/package/Group/GrpData/trngps.m��	*

__sig__107(*��p�����,*NonsplitAbelianSectionl
*��������l� *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.t�!*�$��������������"*

__sig__108��#*��������80*
__sig__111belianSection$*]Determine the maximal p-elementary abelian module with a splitting lift to a bigger quotient.�#%*��$����������������('*��������L*(* NonsplitElementaryAbelianSection-�-.)*����������**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.?4?+*�$��������������B,*

__sig__110,DDD-*��������F.*CDetermine a lift with a p-group, given by its lower central series.�H�H/*��$�������Ĩ�����M1*���������O2*

PGroupSection0Q3*����������4*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.g�g�g5*�$�������ܨ��Ш�m6*

__sig__112�n�n7*��ب����Lp8*NilpotentSectionq�q�q9*��������Ԩ�'__sig__09�8Y	�'

__sig__118>, w�'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. ��`��'�y��R*��t�l���`���'y��R*��d�������H��'

__sig__114���'����������;*4/usr/local/magma/package/Group/GrpMat/Smash/larger.m[	]	<*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�	=*���kR�������[�d)0Return the l-th list of p-modules stored in SQP.	
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	__sig__73��g)���������~)
	__sig__78ector�h)>Delete the collector for split extensions in characteristic p.,i)��������������������Lj)
	__sig__74T|k)���������l)DeleteSplitCollector4m)����������n)+Delete all collectors for split extensions.d#�#o)�����������ĥ�����(p)
	__sig__75�)�)q)��������8,r)BDelete the collector for non-split extensions in characteristic p.|2�2s)�������������ԥ��ȥT:t)
	__sig__76t;�;u)��Х����=v)DeleteNonsplitCollector�>w)��������̥x)/Delete all collectors for non-split extensions.LC�C�Cy)���������������DGz)
	__sig__77$H<H{)��������I|)*Delete the collectors in characteristic p.�M�M})������������������\S)���������U@*����������A*:/usr/local/magma/package/Group/GrpMat/Smash/charpol-test.m�d	�r	B*:/usr/local/magma/package/Group/GrpMat/Smash/extraspecial.m�d	�r	C*:/usr/local/magma/package/Group/GrpMat/Smash/extraspecial.mPe	�r	D*MTrue iff G is known to normalise an extrapecial p-group or symplectic 2-group�E*�$������������F*
__sig__0�G*������PH*IsExtraSpecialNormaliser��I*���������J*7/usr/local/magma/package/Group/GrpMat/Smash/functions.m�r	K*<7/usr/local/magma/package/Group/GrpMat/Smash/functions.m�r	M*���������L�<�,�0�,N*
__sig__0.@.X.O*,�8������1P*
BlockSystem�5<6Q*��������4�R*The block system of M`;S*g��������T�,�H��>T*
__sig__1?�?�?U*,�P�����0Ab*
__sig__4eters CV*FReturn prime and exponent of the extraspecial subgroup normalised by M�FW*�R��������t�d�,�X��IX*
__sig__2JK KY*,�`�����NZ*ExtraSpecialParameters�O[*��������\�\*FReturn prime and exponent of the extraspecial subgroup normalised by MY]*gR��������|�,�p�,^^*
__sig__3_�_�a_*,�x������d`*%Return the extraspecial subgroup of M�ga*��������������,����mc*,�������0od*%Return the extraspecial subgroup of Mqe*g�����������,����tf*
__sig__5w�w$xg*,�������hzh*The tensor factors of M�<�T�i*g/������������,�����j*
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TensorFactors�m*����������n*The tensor factors of G̍���o*�/��������ĩ,���ܑp*
__sig__7����q*,�������ܔr*NReturn the change of basis matrix which exhibits the tensor decomposition of M(�s*g����������ԩ,�ȩ��t*
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DeleteCollectorPZ�)�����������)!Delete all the collectors of SQP.�c�c�)����������������h�)
	__sig__79�jk�)��������m�)-Delete the soluble quotient process variable.�o�)�����������$����ls�)
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DeleteProcesshz�)����������)/Delete the Process and all its child processes.��̂��)�����������<���0����)
	__sig__81����)��8�����4�w*��������̩x*NReturn the change of basis matrix which exhibits the tensor decomposition of GTy*�����������,���{*,��������*
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__sig__28P�F�Q�������FHThe ideal of the appropriate quadratic field defined by the ideal I of a quadratic number field. This function exists only because quadratic fields and number fields are distinct types in MAGMA. Phi is a map from FractionField(Order(I)) to a quadratic field.�&'/F�����������Q�Q��*ZT
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QuadraticIdeal�4jU���������QB�I is an ideal of O_K that is coprime to p. Each of pi and pibar define a map from O_K to Z_p; pi is the image of O.1, and similarly for pibar. (Note that pi and pibar are computed by finding the roots of the charpoly of O.1 over Z_p.)HApAiU�������������H�Q,D�E�R
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__sig__5[�[�[�Q��Q�F)�\^�QNum_and_Denom_Idealsc\ciQ��������GaQ2�The function H_2 defined in Mazur's note. This is rhotilde(J*x(P))/2, where J is the "ideal denominator" of the principal ideal (x(P)) generated by the x-coordinate of P.�o�o_Q�t���������E�Q�E,sAQ��QE)��t(Q��������|E!Q,�Let h(x) = H_2(p^n*x)/p^(2n). Then this returns h(P+Q) + h(P-Q) - 2*h(P) - 2*h(Q), which is an approximation for the height pairing of P and Q.�D�l�Q��tt���������HD�Q�DćQ
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vialityConjectureSQ��N���������S0S�QTE���T
	__sig__11�����T�Q4S�������TTestNontrivialityConjecture�����T��������LEgUj�Let E be a rank two elliptic curve and let P1, P2 be a basis for E(Q). Let P = P1 + P0, where P0 is a point of infinite order in E_D(Q) subset E(Q(sqrt(D))). The value returned is a list of pairs <D, [* H_2(P), H_2(p*P)/p^2, H_2(p^2*P)/p^4, ..., H_2(p^(2*n)*P)/p^(2*n) *]> Some of the H_2 values might equal false, if the point doesn't like in the appropriate subgroup for H_2 to be defined. ������TR��N����������M�Q�U,��S
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	__sig__15� �D4�The images of the ith standard basis vector under the Hecke operators Tj for j<=n. These are computed using sparse methods that don't require computing the full Hecke operator.��8�D
HeckeImagesAll�D����������D^Computes the Hecke module over Z generated by a vector v. The result is returned as a lattice.�D��������������d��<�D
__sig__4���D�������N ����������@���<���U t�A positive integer n such that the Hecke operators T1,...,Tn generate the Hecke algebra as a Z-module. When the character is trivial, the default bound is (k/12)*[SL_2(Z):Gamma_0(N)]. When the character of M is nontrivial, the default bound is twice the above bound; however, it is not known that this bound is large enough in all cases in which the character is nontrivial, so one may wish to increase the bound using SetHeckeBound.[ <�Many computations require a bound n such that T1,...,Tn generate the Hecke algebra as a Z-module. This command allows you to set the bound that is used internally. Setting it too low can result in false answers.
���8` �����������������b ��������g %The same as AtkinLehnerOperator(M,q).f �����������������U-$tThe p-covering group of H; if Exponent is non-zero, then enforce the corresponding exponent law for the p-cover of H
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V-��R�����`�P�T�X-P�\�����TY-Bconstruct subspace for allowable subgroup of P whose quotient is G��Z-���������p�P�d��\-P�l�����]-AllowableSubgroupX�^-��������h�_-.�lengths of orbits and orbit representatives of all k-dimensional subspaces of natural vector space under action of matrix group G defined over prime field$�$`-��R����������P�|�X)a-
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OrbitsOfSpaces<0d-����������e-0/usr/local/magma/package/Group/GrpPC/pgrps/ops.m:�S	�^	f-
0/usr/local/magma/package/Group/GrpPC/pgrps/ops.m<�S	�^	g-JGiven a p-group G and a list of tuples [<G.i, W.i>] for the generators
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 relations which give definitions for the pc-generators. FGGh-RR�R��������������(Ji-
__sig__0KL$Lj-���������Nk-CompleteTupleListPPxPl-����������m-*�Construct the map P->Q of p-groups, given by the list of images of
 the generators of P. D indicates the definitions of pc generators of
 P.\�\n-RR������������������dp-��������tgq-p_hom�hr-����������s-8Construct the inverse homomorphism of h:GrpPC -> GrpPC. o�o�ot-����������ص��̵,su-
__sig__2t�t�tv-��Ե����$xw-InvHom�yx-��������еy-0�Returns the map h^n, the n-th power of the homomorphism h. If the Domain
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 or equal to zero.�<�z-�����������������{-
__sig__3�,�T�|-���������}-PowHom܏~-���������-MGiven a Map as composition of homs, construct the composition map explicitly.��-������������������-
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BuildHom�Ԡ��-����������-5/usr/local/magma/package/Group/GrpPC/pgrps/parallel.m	�^	�-7/usr/local/magma/package/Group/GrpPC/pgrps/parameters.m�^	�-3/usr/local/magma/package/Group/GrpPC/pgrps/shadow.m�]	�^	�-;/usr/local/magma/package/Group/GrpPC/pgrps/stab-of-spaces.m�]	�^	�-
Stabiliser��Ȳ����,HGiven: An irreducible matrix group G, and A specially ordered sequence Q of the vectors in RSpace(G). Return: The image H of the natural permutation representation of the 	semidirect product of G with its natural submodule generated by 	the vector (0, ..., 0, 1)��, ��������,����������,J	Given: An irreducible matrix group G, and A specially ordered indexed set Q of the vectors in RSpace(G). Return: The image H of the natural permutation representation of the 	semidirect product of G with its natural submodule generated by 	the vector (0, ..., 0, 1)H!p!�,������������ ���l%�,
__sig__9'�'�'�, �������)�,?A group from the Isol Group database satisfying the predicate f.@.X.�,���������� ��H7�,
	__sig__10@9�9�, ������(;�-j�Construct descendants of p-group having p-class one larger than G, and order at most p^OrderBound; if Exponent is non-zero, generate descendants of G satisfying this exponent law; if StepSizes is supplied, then construct descendants of order p^(n + s) of a group of order p^n, only for s in StepSizes; by default, all descendants of G are returned; if All = false, only capable ones are returned\t�-h�Generate tree of d-generator p-class <= c p-groups of order at most p^OrderBound; if Exponent is non-zero, generate groups satisfying this exponent law; if StepSizes is supplied, then construct extensions of order p^(n + s) of a group of order p^n only for s in StepSizes; by default, all groups satisfying these properties are returned; if All = false, only capable ones are returned#$�$�-�������'�-
GeneratepGroups�(�-����������-6/usr/local/magma/package/Group/GrpPC/central/central.m80�-</usr/local/magma/package/Group/GrpPC/count-pgrps/backtrack.mg	�r	�-8/usr/local/magma/package/Group/GrpPC/count-pgrps/fixed.mk.m�g	�r	�-8/usr/local/magma/package/Group/GrpPC/count-pgrps/fixed.m>�g	�r	�-4Return number of subspaces of dimension s fixed by g@[email protected]�-����������� �����D�-
__sig__0E�EF�-������tG�-NumberOfFixedSpaces�H�H�-����������-4Return number of subspaces of dimension s fixed by gOlO�-�����������(���PU�-
__sig__1WtW�W�-�$������Y�-:/usr/local/magma/package/Group/GrpPC/count-pgrps/classes.m�j	�r	�-:/usr/local/magma/package/Group/GrpPC/count-pgrps/classes.m8k	�r	�-HReturn representatives of conjugacy classes of GL (n, q) and their sizesi�i�i�-��R��������@�0�4��n�-
__sig__0o�op�-0�<������q�-ConjugacyClassesGLPshs�-��������8��-7/usr/local/magma/package/Group/GrpPC/count-pgrps/auto.m�r	�-</usr/local/magma/package/Group/GrpPC/count-pgrps/subgroups.mm	�r	�-</usr/local/magma/package/Group/GrpPC/count-pgrps/subgroups.mm	�r	�-Return order of GL (n, p)P����-�����������d�T�X����-
__sig__0�x����-T�`��������-��������\��-ClassTwoAbelianPGrouproup�-&xOrder of automorphism group of abelian p-group G where a = [a[1], a[2], a[3], ...] and G = C_a[1] x C_a[2] x C_a[3] ... ���$��-R���������|�T�p����-
__sig__1�p����-T�x�����Ȣ�-#OrderAutomorphismGroupAbelianPGroup�(��-��������t��-*�Return the number of subgroups of each non-trivial order in the abelian p-group G where a = [a[1], a[2], ...] and G = C_a[1] x C_a[2] x ...X�x��-RR����������T���IJ�,IsolGroupSatisfying�>�>�,����������,
	__sig__13atisfying�,@The groups in the Isol Group database satisfying the predicate fE�E�E�,R��������0� �$��H�,
	__sig__11J$J�, �,�����|M�,IsolGroupsSatisfyingN�N�,��������(��,KA group of degree n from the Isol Group database satisfying the predicate f,XdX�,����������H� �<��\�,
	__sig__12�^�^�, �D�����d�,IsolGroupOfDegreeSatisfying�fg�,��������@��,	__sig__16FieldSatisfying�,KThe groups of degree n in the Isol Group database satifying the predicate f4pLp�,�R��������`� �T�0t�, �\�����@v�,IsolGroupsOfDegreeSatisfyingz(z�,��������X��,VA group of degree n over GF(p) from the Isol Group database satisfying the predicate fp��,�����������x� �l�X��,
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�-�����������-8/usr/local/magma/package/Group/GrpPC/count-pgrps/count.m�y	��	�-8/usr/local/magma/package/Group/GrpPC/count-pgrps/count.mPz	��	�-&xCount all p-class 2 d-generator groups of order p^(d + s); if Exponent is true, count those groups which have exponent p���-������������ȷ��������-
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ClassTwo!�!�!�-�����������-(�Count all p-class 2 d-generator groups of order p^(d + s) for s in Step; if Exponent is true, count those groups which have exponent p�*�-R��R��������طз��ķ�1�-
__sig__15�5<6�-��̷����|8�- eCount all p-class 2 d-generator groups; if Exponent is true, count those groups which have exponent p`=�-��R�����������Է[email protected]�-
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ElementSequence�I.����������.?generators for Ext (G/G', U) as cocyclic matrices; U is abelianxP�PQ.��RR����������W.
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.����������..�return invariants of abelian group Z^2 (G, U) where U is an abelian group; Ext and Hom are generators for Ext (G/G', U) and Hom (H_2 (G), U) respectivelyt~�~.RRR��������D��8���.
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__sig__6ycles�.Ireturn representative cocycles from G to (abelian) U as cocyclic matricesL�d�.RR��R��������T��H��.
__sig__5���ܑ.�P�����L�.RepresentativeCocycles,�.��������L�.=central extensions of (abelian) U by G determined by CocyclesX�.����R��������l��`��.�h�����|�.CentralExtensions�,�.��������d�&.
__sig__8rocess,�.9set up process for central extensions of (abelian) U by G���.�������������x�б .
__sig__7���ȲG.AutGpSGp�, IsolGroupOfDegreeFieldSatisfying�,�\��,��������p��,YThe groups of degree n over GF(p) from the Isol Group database satisfying the predicate fؕ��,��R���������� ������,
	__sig__15�H��, �������0��,!IsolGroupsOfDegreeFieldSatisfying���,�����������,9Process to search through all Isol Groups in the database4�L��,����������� ���$��, �������|�X.
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[.��������\�\.BA finitely presented group F isomorphic to A, and isomorphism F->A ].vy�������|�T�p��^.
__sig__1��_.T�x�����<`.\A finitely presented group O isomorphic to A/Inner(A), and natural epimorphism A`FpGroup->O �a.vy���������T����b.
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OuterFPGroup#�#e.����������f. _Decide whether a is inner and if so also return a corresponding conjugating element of A`Group 8,�,D-g.!$���������T����5h.
__sig__37�7�7i.T�������T:j. _Decide whether a is inner and if so also return a corresponding conjugating element of A`Group t>�>�>k.!$���������T���lAl.
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__sig__6iesH<Hp.HThe characteristic series of A`Group (from top down) used to compute A. L8L|Mq.vR��������̹T���0Qr.
__sig__5SDTtTs.T�ȹ����4Wt.CharacteristicSeriesYLYu.��������Ĺv.BA finitely presented group F isomorphic to A, and isomorphism F->A�c�cw.v���������T�ع�hy.T������tkz.8/usr/local/magma/package/Group/GrpPerm/max/complements.mnȔ	X�	{.8/usr/local/magma/package/Group/GrpPerm/max/complements.mq �	X�	|.B�Given a finite permutation group G, with normal subgroup M, if M has a complement in G return a list of representatives of the conjugacy classes of complements in G. If M does not have a complement in G, the empty sequence is returned����}.��R��������,�������~.
__sig__0� �8�.����������..�Given a finite permutation group G, with normal subgroup M, if M has a complement in G return true, otherwise false. A single complement is also returned.���.��$����������ؓ�.
__sig__1�@�X��.���������.NGiven a finite permutation group G, with normal subgroups N and M, such that N <= M, this returns a list L of subgroups of G, all satisfying N < L[i] < M, such that L[1]/N and L[i+1]/L[i] are elementary abelian or direct products of nonabelian simple groups for all i, and L[#L] = Mp����.���R�����������Ȩ�.
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IsolProcess,�D��,�����������.JGiven a finite permutation group G, with normal subgroups N and M, such that N < M, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of complements in G/N. If M/N does not have a complement in G/N, the empty sequence is returnedT�.���R��������4��(���.�0�����0�.JGiven a finite permutation group G, with normal subgroups N and M, such that N < M, if M/N has a complement in G/N return a list of representatives of the conjugacy classes of complements in G/N. If M/N does not have a complement in G/N, the empty sequence is returned ` �.���R��������D��8��$�.
__sig__4&�&'�.�@�����)�.D�Given a finite permutation group G, with soluble normal subgroup M, if M has a supplement in G return a list of representatives of the conjugacy classes of supplements in G. If M does not have a supplement in G, the empty sequence is returned:(:�.��R��������|�T��H�`=�.
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SupplementsA0A�.��������L��.0�Given a finite permutation group G, with soluble normal subgroup M, if M has a supplement in G return true, otherwise false. A single supplement is also returned.J(J�.��$�������l��`�pO�.
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HasSupplement�V�.��������d��.NGiven 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 supplement 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 supplement in G/N, the empty sequence is returnedxk�k�k�.���R�����������x��o�.
__sig__7p�pq�.��������r�.NGiven 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 supplement 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 supplement in G/N, the empty sequence is returned���t��.���R�������������\��.
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Supplement��̍�.�����������.4/usr/local/magma/package/Group/GrpPerm/max/maxcomp.m�	X�	�.4/usr/local/magma/package/Group/GrpPerm/max/maxcomp.m�	X�	�.2�G should be a permutation group with trivial soluble radical (i.e. a TF-group). The maximal subgroups of G are computed, including G. A list subgroup records is returned. �0��.�R��������������<��.
__sig__0�D�\��.�����������.MaximalSubgroupsTF�,��.�����������.For internal use���ȭ�.��R��������̺����h��.
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psnsion�,WProcess to search through all Isol Groups in the database with either degree equal to n�,����������� ����,
	__sig__17�, ��������,IsolProcessOfDegree�,�����������,\Process to search through all Isol Groups in the database with degree in the range specified�,
	__sig__18H/D�Given a sequence of complexes of modules over the same algebra, the function returns the sequence of complexes obtained by taking Zero extensions of the elements of S, if necessary until all of the elements of the sequence have the same degrees.�M/��������$�N/D�Given a complex C in degrees from a to b and a complex D in degrees c to d, the function returns the two complexes in degrees max(a,c+n) to min(b,d+n) and from max(a-n,c) to min(b-n,d) obtained from C and D by zero extensions when necessary.��O/����������L�D�d�8��P/
	__sig__23� � Q/d�@������"R/6�Given complexes C and D over the same algebra, the function returns the complexes obtained by taking Zero extensions of C and D, if necessary so that both complexes have the same degrees. ,8,S/����������T�d�H��4T/
	__sig__24\7t7U/d�P�����$:V/@�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.\BW/����������d�d�X��FX/
	__sig__25\GtGY/d�`������HZ/LeftExactExtension<JTJ[/��������\�\/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.[�[]/����������|�d�p�Xc^/
	__sig__26eHe_/d�x������g`/RightExactExtension�i�ia/��������t�b/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.x}c/������������d���H�d/
	__sig__27����e/d��������f/
ExactExtension��g/����������h/=Returns the complex C with n terms removed from the left end.ȍi/������������d�����j/
	__sig__28��Ȓk/d���������l/>Returns the complex C with n terms removed from the right end.�m/������������d���|�n/
	__sig__29�,�o/d���������p/)The sequence of the terms of the complex.����q/�R��������̽d���x�r/
	__sig__30���s/d�Ƚ�����t//The list of the boundary maps in the complex C.,�D�\�u/����������ܽd�нd�v/
	__sig__31L�d�w/d�ؽ�������1-/usr/local/magma/package/Lattice/Lattice.specp�!.��������
".CentralExtensionProcess#.��������|�$.-true iff central extension process P is empty%.$��������T������'.�������<(.8construct next central extension determined by process P��).������������������*.
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NextExtension�"-.����������..1/usr/local/magma/package/Group/GrpPerm/dtgroups.m��	0�	/.1/usr/local/magma/package/Group/GrpPerm/dtgroups.m�	0�	9.ClassActionNew7890.$Identify a doubly-transitive group g5861.�/��������̸����$;2.
__sig__0<,<D<3.��ȸ����>4. TwoTransitiveGroupIdentification?�?�?5.��������ĸ6.7/usr/local/magma/package/Group/GrpPerm/aut/attributes.m0�	7.2/usr/local/magma/package/Group/GrpPerm/aut/autgp.mes.m0�	;.
PCGrouproup7:.InverseClassActionKK?<1/usr/local/magma/package/RepThry/ModGrp/declare.m><.2/usr/local/magma/package/Group/GrpPerm/aut/autgp.m`�	0�	=./The automorphism group of a permutation group G�W,XdX>.�v��������������\?.
__sig__0^�^�^@.�������dA.4/usr/local/magma/package/Group/GrpPerm/aut/autsdgp.m�	0�	B.4/usr/local/magma/package/Group/GrpPerm/aut/autsdgp.m�	0�	C.BThe automorphism group of a semidirect product group G=NH fixing N�o�oD.���v�����������hsE.
__sig__0t�t�tF.�������xH.���������I.6/usr/local/magma/package/Group/GrpPerm/aut/backtrack.m	0�	J.6/usr/local/magma/package/Group/GrpPerm/aut/extendaut.m	0�	K.4/usr/local/magma/package/Group/GrpPerm/aut/isomgps.m�	0�	L.4/usr/local/magma/package/Group/GrpPerm/aut/isomgps.m�	0�	M.1True if permutation groups G and H are isomorphic@�X�N.��$���������@�0�4�8�O.
__sig__0�p���P.0�<�����X�Q.3/usr/local/magma/package/Group/GrpPerm/aut/oddfns.m�	0�	R.4/usr/local/magma/package/Group/GrpPerm/aut/radquot.m�	0�	S.9/usr/local/magma/package/Group/GrpPerm/aut/refineseries.m	��	0�	T.4/usr/local/magma/package/Group/GrpPerm/aut/grpauto.m�	0�	U.4/usr/local/magma/package/Group/GrpPerm/aut/grpauto.m�	0�	V.D�Find a faithful permutation representation of A on union of conjugacy classes of G. Return the isomorphism phi from A to a permutation group P, the permutation group P itself (acting on 1,..,n) and the union of classes on which it actually acts. IJW.v��R����d�T�X�ĵ�-;/usr/local/magma/package/Group/GrpPC/pgrps/stab-of-spaces.m0�	��	�- _return indexed set of elements of G listed in the order used by ExtGenerators and HomGenerators�-����RR������������-,�Subgroup of GL(d, F) which fixes sequence of subspaces of natural vector space; also return generators for largest unipotent subgroup of stabiliserT|�-$�4�������-StabiliserOfSpaces4�-��������0��-5/usr/local/magma/package/Group/GrpPC/pgrps/standard.m	��	�-5/usr/local/magma/package/Group/GrpPC/pgrps/standard.m	��	�-
Standard)l)�)�-JP is p-covering group of H; M is relative multiplicator; K is class c quotient; H is class c - 1 quotient; if Relative is non-negative, then it is the relative step size; otherwise step size is log of #K div #H; return matrix of (relative) allowable subgroup from P to K\;�-�������������\�H�P��>�-
__sig__0?�?�?�-H�X�����,A�-SubgroupToMatrixC4CLC�-��������T��-
__sig__2ionFG�-8return standard presentation H for G and map from G to HI�I�I�-���������t�H�h��N�-
__sig__1OP4P�-H�p������S�-StandardPresentationVdV�-��������l��-Yreturn true and a map from G to H if the p-groups G and H are isomorphic; otherwise false�^�^�-��$�������8���H���g�-H��������h�-5/usr/local/magma/package/Group/GrpPC/pgrps/generate.m	��	�-5/usr/local/magma/package/Group/GrpPC/pgrps/generate.m	��	�-GeneratepGroups,qdq�q�-��������������ȅ�/�-*�return s-step definition sets and offsets of dimension n in vector space of dimension m, where we must retain the first Fixed basis elements�H��-
__sig__0������-����������-
DefinitionSets���-�����������-j�Construct descendants of p-group having p-class at most ClassBound, and order at most p^OrderBound; if Exponent is non-zero, generate descendants of G satisfying this exponent law; if StepSizes is supplied, then construct descendants of order p^(n + s) of a group of order p^n, only for s in StepSizes; by default, all descendants of G are returned; if All = false, only capable ones are returned����-��R�������������H��-
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Descendants@�X��-�����������-�R��������ض��̶��-
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__sig__7(�/��������,�/&~Returns the pushout of the diagram [ M1 <-- fc1 -- D --- fc2 --> M1 ] as an AModule together with homomorphisms from M1 and M2�
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__sig__8Tl�/����������/�����������/&}Returns the pullback of the diagram [ M1 -- fc1 --> N <-- fc2 -- M2 ] as an AModule together with homomorphisms to M1 and M2.�/��������ľ����� �/
__sig__9"T"|"�/���������$�/$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical.�*�*�/�"��������Ծ��ȾP1�/
	__sig__10\5�5�/��о�����7�/IrreducibleModulel:�:�/��������̾�/$rThe nth irreducible module of the algebra. The module is the quotient of the nth projective module by its radical.�@�@�/�"������������lE�/
	__sig__11�F�F�/��������G�/
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	__sig__13ionN�/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.LY�/#��������������X^�/
	__sig__12�a�b�/�������He�/"nTht algebra $A$ as a right module over itself. The module is the direct sum of the projectives modules of $A$.l�/"������������p�/��������q�/RightRegularModulePshs�/����������/	__sig__16iveModules�y�/RThe sequence of the dimensions of the projective modules of the basic algebra $A$.�����/"R��������,��� ����/
	__sig__14 �8��/��(��������/DimensionsOfProjectiveModules���/��������$��/QThe sequence of the dimensions of the injective modules of the basic algebra $A$.���/"R��������D���8����/
	__sig__15l����/��@�����ę�/DimensionsOfInjectiveModules����/��������<��/V7True 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.����ĭ�/$R������@�\���P�d��/��X��������/4Returns the identity (n x n)-matrix over the ring R.����/������������l���`�й�/�.��Ⱥ����,
�.

MaxSubsTF2$T�.��������ĺ�.For internal use���.$��R�����������غl�.
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MaxSubsTF34�.��������ܺ�.2�Find maximal subgroups of G modulo the soluble normal subgroup N. Use MaximalSubgroupsTF on the radical quotient where necessary. A list subgroup records is returned. �!�!"�.��R�������������l&�.
__sig__3(t(�(�.���������)�.MaximalSubgroupsH ,8,�.�����������.5/usr/local/magma/package/Group/GrpPerm/max/identify.m	��	�.5/usr/local/magma/package/Group/GrpPerm/max/identify.m	��	�.
__sig__0pleGroup�.1Look up G in the database of almost simple groups�>?�.���������,����\B�.
__sig__0C�CD�.�������E�.IdentifyAlmostSimpleGroup\GtG�.����������.1Look up G in the database of almost simple groups�M�M�.���������4��(�dR�.
__sig__1T�T�T�.�0������W�.3/usr/local/magma/package/Group/GrpPerm/max/oddfns.m �	��	�.2/usr/local/magma/package/Group/GrpPerm/primitive.mp�	��	�.2/usr/local/magma/package/Group/GrpPerm/primitive.m��	��	�. fGiven an affine group G, compute the socle quotient as a subgroup of GL(n, p), where Degree(G) eq p^n.l�.���������P�@�D�p�.@�L������q�.
MatrixQuotient(s�.��������H��;%/usr/local/magma/package/Opt/Opt.specpec�.0/usr/local/magma/package/HomAlg/AlgBas/complex.m���	��	�.B0/usr/local/magma/package/HomAlg/AlgBas/complex.m� �	��	�.WCreate the complex given by the list L of maps and such that the last term has degree d�@�X��.�����������t�d�h���.
__sig__0��(��.d�p�����؏�.<�The splice of the complex C with the complex D along the map f from the last term of C to the first term of D. the degree of the last term of the splice is the same as the degree of the last term of the complex D.$��.��������������d�x����.
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__sig__3mplexh��.0The sequence of homology groups of the complex C������.�R����������d���P��.
__sig__2�ܬ��.d�������,��.HomologyOfChainComplex̱�.�����������.0The sequence of homology groups of the complex C���ĵ�.d�������h��.Homology�����и��/
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�/@True if the map f is a homomorphism of modules over the algebra. �/�%������������x���/
	__sig__18��/��������x�/IsModuleHomomorphism��/��������|��/-/usr/local/magma/package/HomAlg/AlgBas/attr.m	
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	�/OppositeAlgebraH#h#�#�/CompactProjectiveResolution�%p&�/CompactInjectiveResolution))�/�/Dual+�+�/HomologyOfChainComplexh��/1/usr/local/magma/package/HomAlg/AlgBas/chainmap.m`5�5�/1/usr/local/magma/package/HomAlg/AlgBas/chainmap.m:(:�/6�Given a projective resolutions P for a simple module S over a basic algebra A, the function returns the chain maps in compact form of the minimal generators for the cohomology Ext*(S,S).�@�@�///��������п��Ŀ0E�/
__sig__0FpF�F�/��̿�����G�/CohomologyRingGeneratorsIPIpI�/��������ȿ�/"gGiven the generators for cohomolgy, the function returns the list of degrees of the minimal generators.`SxS�S�//R�����������ܿY�/��������Z�/DegreesOfGenerators�\�\�/���������;0__sig__5ionutionp�e�/V: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 in compact form of the minimal generators for the cohomology Ext*(S,T) as a right module over the cohomology ring Ext*(T,T).t4t�/////�������������,z�/
__sig__2~t~�~0��������L�0CohomologyRightModuleGenerators��0����������0V9Given 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 in compact form of the minimal generators for the cohomology Ext*(S,T) as a left module over the cohomology ring Ext*(S,S).��̒0////������������ؖ0
__sig__3� �8�0���������0CohomologyLeftModuleGenerators��0���������	0IReturns the degrees of the chain maps of the generators of the cohomology�,�
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__sig__5��@�0��D������0CohomologyGeneratorToChainMap\�0
__sig__6������.�R������������d�����.�����������."hThe homology group in degree n of the complex C as an A-module, together with the associated epimorphismXp�.����������Ļd�����.
__sig__4���.d��������.9The dimension of the n-th homology group of the complex C`x�.��R��������Իd�Ȼ�!�.
__sig__5"�"#�.d�л����4%�.DimensionOfHomology�'�'�.��������̻/
__sig__7ogy+,�.FThe sequence of the dimensions of the homology groups of the complex C�1�.�R���������d��(:�.
__sig__6;H;`;�.d�������<�.DimensionsOfHomology>�>�.����������.$oThe subcomplex of C generated by S and the inclusion of S in C. S is a sequence of submodules of the terms of C�E�E�E�.�������������d����H/d������XJ/

Subcomplex<L�M/����������/NThe 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.�_�a/�����������d���g/
__sig__8hipi/d������l/X?Given a chain complex C in degrees a to a-t+1 and a sequence S = [s_1 .. s_t], the function creates the minimal chain comples whose term in degree a-i+1 is a subcomplex generated by s_i random elements of the term in degree a-i+1 of C. The function also returns the chain map that is the inclusion of subcomplex into C.|}�}L~	/R���������,�d� ���
/
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/��������$�/?The quotient complex Q of C by D and the projection of C onto Q,�D�\�/�����������T�D�d�8�<�/
	__sig__10t���/d�@�����\�/
QuotientComplex(�/��������<�/9The quotient complex of C by the sequence S of submodules����/�����������d�\�d�P�|�/
	__sig__11Ԥ��/d�X�����\�/P The quotient of C by the subcomplex generated by L. The list L may be either a list of sequences of element of the terms of C or a list of submodules of the terms of C. There must be one element of L for each term of C. The function returns both the quotient complex and the quotient map.�P�h�/����������l�d�`�|�/
	__sig__12����/Pushout�83*�������������4���(;3HackobjPrintSymGenLoc@<3����������=3
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__sig__7�@34��������A3HackobjPrintSymGen�B3����������D3��$�������� ��4��<E3
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	__sig__11tive�3K3A representative lattice of G.�=L3������������8�4�,��@M3
	__sig__108BN34�4������CO3InternalRepresentative(FP3��������0�Q3!True if and only if G is a genus. MR3�$��������P�4�D�TMT34�L�����RU3IsGenus,RV3��������H��3SformsesW3(True if and only if G is a spinor genus.[8[X3�$��������h�4�\�aY3
	__sig__12 aZ34�d�����8a[3

IsSpinorGenusLa\3��������`�]3&The primes of the local genus symbols.�a^3������������4�t��i_3
	__sig__13�i`34�|������ra3The prime of the genus symbol.$ub3������������4����zc3
	__sig__14�zd34�������{e3The p-adic determinant of G.{f3��������������4���P{g3
	__sig__15d{h34�������|{i3The determinant of G.�{j3������������4����{k3
	__sig__16�{l34��������{m3The rank of the genus.|n3��������������4���@|o3
	__sig__17�p34��������q3The rank of the genus.4�r3��������������4���ԟs3
	__sig__18 �t34�������8�u3The rank of the genus.P�v3����������x���4��� �w3
	__sig__194�x34�������L�y3The rank of the genus.d�z3����������h���4�����{3
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	__sig__22tiveԴ}3@A canonical representative for the p-adic genus, in Jordan form.���,0V7Given the projective resolutions P and Q of two modules M and N and the cohomology generators in compact form C of the cohomology module, Ext_B^*(M,N), the function returns the chain map from P to Q that lifts the nth generator of the of the cohomology ring and has degree equal to the degree of that generator.@Xp0��������@�0P$Given the projective resolution P of a module and the cohomology generators in compact form C of the cohomology ring of that module, the function returns the chain map from P to P that lifts the nth generator of the of the cohomology ring and has degree equal to the degree of that generator.!�!0�/����������`���T��%0��\������(00/usr/local/magma/package/HomAlg/AlgBas/compact.m*�0	�=	00/usr/local/magma/package/HomAlg/AlgBas/compact.m01	�=	0MThe sequence of sequences of dimensions of the homology groups been computed.|80R��������x�h�l��<0
__sig__0=�=>0h�t�����h?0SimpleHomologyDimensions@�@A 0��������p�!0,�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.8IPIpI"0R����������h���PN#0
__sig__1O�O�O$0h�������`S%0SimpleCohomologyDimensions@VhV&0����������'0��Given a module M over a basic algebra and a natural number n the function computes a projective resolution for M out to n steps. The function returns the resolution in compact form together with the augmentation map (P_0 -> M). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (P_{i+1} -> P_i) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable projective modules making up P_{i+1} in the indecomposable projective modules making up P_i and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in P_{i+1} and in P_i.4�L�(0�/����������h�����)0
__sig__2���*0h�������D�+0CompactProjectiveResolutionL�d�,0����������-0�yGiven a module M over a basic algebra and a natural number n the function computes an injective resolution for M out to n steps. The function returns the resolution in compact form together with the coaugmentation map (M -> I_0). The compact form of the resolution is a list of the the minimal pieces of information needed to reconstruct the boundary maps in the resolution. That is the boundary map (I_i -> I_{i-1}) is recorded as a tuple consisting of a matrix whose entries are the images of the generators for indecomposable injective modules making up I_i in the indecomposable injective modules making up I_{i-1} and two lists of integers givin the number of indecomposable projective modules of each isomorphism class in I_i and in I_{i-1}. The actual return of the function is the compact projective resolution of the dual module of M over the opposite algebra of the algebra of M. �8�.0�/����������h�����/0
__sig__3�Ը �00h��������10CompactInjectiveResolution�4�x/OTrue if the complex consists of a short exact sequence along 	 with other terms(Xy/�$����������d���z/
	__sig__32��{/d�������|/IsShortExactSequence}/���������/&1/usr/local/magma/package/HomAlg/AlgBas/algebras.mX?	�M	�/��������������<�/
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ismesi0BCreate the chain map from C to D all of whose module maps are zeroT|j0�����������������|k0
__sig__0d�l0���������m0
ZeroChainMap,n0����������o01Checks to see if we can take kernel and cokernel.H%h%p0�$���������������)q0
__sig__1+�+,r0���������.s0IsProperChainMap1�1�1t0����������u00�True if the list of maps L from the terms of complex C to the terms of D is a chain complex of degree n, i. e. it has the right lenghth and the diagram commutes>�>?v0����$����������������\Bw0
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IsChainMap�FGz0����������{0-True if the chain map f really is a chain map�K|0�$��������������tP}0
__sig__3StS�S~0��������dV0QThe first and last degree of the domain of f on which the chain map f is defined.d[�[�0��������������Xc�0
__sig__4deHe�0��������g�0DefinedInDegreesi�i�i�0����������0The module maps of fo,o�0���������� �����r�0
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ModuleMapsXyxy�0����������08�The kernel complex of f and the chain map of the inclusion. In the event that the chain map is not defined on a particular term of the Domain then the entire term is assumed to be in the kernel.����0���������8���,�`��0
__sig__6������0��4��������0OThe cokernel complex of f and the chain map of the projection onto the quotientH�p����0���������H���<�ԗ�0
__sig__7���ę�0��D�����T��0>�The image complex I of f and the inclusion of I into the codomain and the projection of the domain onto I. If the chain map has no defined image into a particular term of the codomain then the image is assumed to be zero����0��������X���L���0
__sig__8�����0��T��������0%The sum of the two chain maps f and g���0������������h���\����0
__sig__9�ܵ���0��d��������0/The product of the scalar s and the chain map f����к�0
	__sig__10�0��0��t�����(��/
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__sig__2gBasT)�/The Jacobson radical of Md+�+�/��������$�����1�/
__sig__16�6�6�/�� ������9�/JacobsonRadicalAlgBas$;�/����������/>�Given an element x in a module over a basic algebra, creates the homomorphism from the nth projective module for the algebra to the module with the property that the idempotent e of the projective module maps to x*e�E�E�E�/����������\�<���0��H�/��8�����TJ�/LiftHomomorphismM�M�M�/��������4��/NGiven a sequence S = [s_1,s_2, ... ], the function returns a projective module which is the direct sum of s_1 copies of the first projective of the algebra A, s_2 copies of the second, etc. It also returns the sequences of inclusions and projections onto the incecomposable projecitves.�b�/R"RR����T���H��g�/
__sig__3i�i�i�/��P�����Tl�/``Given an sequence X of elements in a module over a basic algebra and a sequence N of nonnegative integers, the function creates the homomorphism from the projective module that is a direct sum of N[i] copies of the ith projective module for the algebra to the module of X with the property that the idempotent e of the projective module maps to X[i]*e.������/RR���������d���X�p��/
__sig__4�����/��`��������/^]The projective cover of M given as the projective module P, the surjective homomorphism of P onto M, the sequences of inclusion and projection homomorphism of P from and to its indecomposable direct summands and the isomorphism type of P in the form of a list of the number of copies of the projective modules of algebra of each type that make up P.��/�RRRt���h����/
__sig__5���D��/��p�����,��/
ProjectiveCover��/��������l��/0�Returns true if the module is projective and returns a list of how many projective modules of each type are direct summands of the projective cover of the module.(�P��/$R���������������/
__sig__6�����/��������4��/.�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.��/$R��������������0��0�������������x���l���0-The composition of the two chain maps f and g��0�����������������|���0
	__sig__11���0���������0;True if and only if the chain map f is zero in every degreeD\�0�$����������������0
	__sig__12���0���������0CTrue if and only if the chain map f is an injection in every degreeD \ �0�$���������������$�0
	__sig__13�&'�0��������)�0CTrue if and only if the chain map f is a surjection in every degreel-�-�0�$��������������86�0
	__sig__14�7�7�0���������:�0ETrue if and only if the chain map f is an isomorphism in every degree�=�0�$���������������@�0
	__sig__15�A\B�0��������DD�0
__sig__3ulemology�0*�True if the sequence of chain complexes, 0 -> Domain(f) -> Domain(g) -> Codomain(g) -> 0, where the internal maps are f and g, is exact.JKK�0��$��������������4P�0
	__sig__16�R\S�0���������U�0DThe homomorphism induced on homology by the chain map f in degree n.Y�Y�0������������������_�0
	__sig__17�cd�0���������f�0InducedMapOnHomologyh�h�0�����������0&wThe connecting homomorphism in degree n of the short exact sequence of chain complexes given by the chain maps f and g.,qdq�q�0�����������������v�0
	__sig__18y0y�0��������x}�0ConnectingHomomorphism���0�����������0:�The long exact sequence on homology for the exact sequence of complexes given by the chain maps f and g as a chain complex with the homolgy group in degree i for the Cokernel of C appearing in degree 3i.�0��0���������������؏�0
	__sig__19P����0����������0LongExactSequenceOnHomology�(��0����������0./usr/local/magma/package/HomAlg/AlgBas/opalg.m	xo	�0./usr/local/magma/package/HomAlg/AlgBas/opalg.m	xo	�0�����������8�(�,����0
__sig__0��(��0(�4��������0PathTreeCyclicModule�ا�0��������0��0,�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.IJ�0""��������P�(�D�ĵ�0
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OppositeAlgebra���0��������H��0"���������h�(�\�������20����������30F�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.�40�����������h����50
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