Algebraic Geometry

• Schemes

  • Machinery has been provided for computing the sheaf of differentials and the tangent bundle sheaf for an ordinary projective scheme.

  • The ability to determine the multiplicity for singular points has been extended from plane curves to general schemes. Computing the tangent line has been extended from plane curves to general curves.

• Algebraic Surfaces

  • Code has been developed for the parameterisation of singular Del Pezzo surfaces of degrees 3 and 4 by Michael Harrison and Josef Schicho. The non-singular case was made available in a previous release so that all degree 3 and 4 Del Pezzo surfaces are now handled.

• Toric Geometry

  • The toric geometry machinery is more tightly coupled with the existing schemes code. For example, the scheme returned by ProjectiveSpace can be passed directly to "toric" intrinsics such as Fan.

  • As part of this tighter integration, Dimension, IsProjective, IsAffine, and EmptySubscheme now work for a larger collection of subschemes, such as the case when the ambient toric variety is non-Q-factorial.

  • The scrolls type PrjScrl has been removed. The constructors RuledSurface and RationalScroll now return type TorVar. This allows a greater range of computations to be performed using these surfaces.

  • Basic quotient singularity analysis is now supported and checking for singularities has been made significantly faster. In the case when the variety is a (fake) weighted projective space it is often sufficient to consider the Cox data, and the fan need not be constructed.

  • A new deterministic algorithm for the resolution of singularities which delivers greatly improved performance has been developed. Similarly, deterministic algorithms for computing a Q-factorial refinement, a Q-factorial terminal refinement, and a Q-factorial canonical refinement of a fan have been added.

  • Databases of the classification of smooth toric Fano varieties up to dimension 8, and of the canonical Fano varieties in dimension 3, are provided.

  • See also Convex Polytopes and Polyhedra.

Arithmetic Fields

• Global Fields

  • The fast algorithm used to compute the maximal order of a Kummer extension has been extended to produce the maximal order of a radical extension. It is much faster than the Zassenhaus Round 2–4 algorithms.

  • It is now possible to calculate the subfields of any global field that is defined by a single polynomial (algorithm of Klueners and van Hoeij).

• Galois Theory

  • Galois groups are now supported for reducible polynomials over number fields (extending the case for polynomials over Z)

  • Galois groups are now supported for relative function fields in characteristic p.

Arithmetic Geometry

• Conics

  • The machinery for solving conics over number fields has been extensively rewritten resulting in much improved performance especially for conics over larger degree fields.

  • Other important infrastructure, such as Simon-minimization, has been added.

• Elliptic Curves Over Finite Fields

  • The code for computing discrete logarithms for elliptic curves over finite fields has been extensively revised leading to substantial speedups.

• Elliptic Curves Over ℚ

  • Code developed by Tom Fisher for performing 8-descent, 6-descent and 12-descent on an elliptic curve has been included.

  • Improvements have been made to 2-descent, in particular large power products are now successfully handled.

  • The routine for "saturating" Mordell-Weil groups has been revised, resulting in speedups by a large factor especially for curves of high rank. (Improved discrete logs contributed part of the speedup).

  • The computation of p-adic L-series in the case of good ordinary reduction has been added. This can also be applied to quadratic twists.

• Elliptic Curves Over Number Fields

  • Periods of elliptic curves over a number field can now be calculated.

  • Analytic information related to the Birch–Swinnerton-Dyer conjecture for elliptic curves over number fields may now be computed: ConjecturalRegulator, ConjecturalSha and AnalyticRank.

• Models of genus one curves

  • Systematic code for testing local solubility of models of degree 2,3 and 4 is included.

  • A general purpose Minimisation routine for genus one normal curves of higher degree is included.

  • A careful Minimisation routine for degree two models over number fields is included.

• Hyperelliptic Curves

  • The 2-Selmer groups of hyperelliptic Jacobians can now be computed in more cases; in particular, for all genus 2 curves over number fields.

  • It is now possible to construct L-series for general hyperelliptic curves defined over the rationals. This relies on the new machinery for regular models. Magma is the only computer algebra system that supports the computation of L-series for curves in this generality.

  • Given the Jacobian of a genus 2 curve, code has been developed by Nils Bruin to compute all (2,2)-isogenous abelian surfaces.

• Regular Models

  • Computation of regular models of (arithmetic surfaces associated to) curves over ℚ or number fields has undergone continued development. Plane curves in (weighted) projective space, in particular hyperelliptic curves, are now handled. The main routine has been made more effective, and some new features of regular models are provided.

Modular Arithmetic Geometry

• Modular Forms and Modular Symbols

  • Major speedups have been achieved in key routines such as NewSubspace and NewformDecomposition, especially for spaces of large dimension.

• Hilbert Modular Forms

  • The implementation of Dembélé's algorithm (the "definite method") has been massively improved in a variety of ways.

  • The case of parallel weight 2 now has separate code.

  • Both phases of the "precomputation" are now considerably faster.

  • Hecke eigenvalues can be obtained without computing full Hecke operators.

  • Much better linear algebra techniques are used in routines such as NewformDecomposition, making it feasible to handle spaces of large dimension.

  • A Hecke-invariant inner product is available in some cases.

Associative Algebras

• Quaternion Algebras

  • Computation of right ideal classes and order classes (for algebras over number fields) has been improved using new algorithmic ideas, resulting in massive speedups.

  • Quaternion algebras over function fields F_q[t], with q odd, have been implemented for the first time by Markus Kirschmer. The standard arithmetic operations are supported (maximal orders, ideal classes, and so on).

• Basic Algebras

  • V2.17 includes machinery for constructing the basic algebra of a modular group algebra. The method has been successfully applied to groups having order up to several million.

  • As a variant on the above feature, it is now possible to compute the basic algebra associated with any p-block of a modular group algebra.

  • The algorithm to construct a basic algebra from a matrix algebra now detects a diagonal block structure in the input, and exploits that structure to greatly reduce the cost of matrix multiplication.

Combinatorial Theory

• Graph Theory

  • The code for graphs with edge labels has been substantially revised to properly handle the labels in all circumstances.

• Hadamard Matrices

  • The database of Hadamard matrices has been expanded. New matrices have been added for degrees 36 (1), 48 (60) and 60 (1759). The additional matrices were supplied by Dragomir Djokovic.

Convex Polytopes and Polyhedra

• Cones, Polytopes, and Polyhedra

  • A vast number of improvements have been made throughout the package for polytopes and polyhedra after the initial release of the package in V2.16. The functionality has been expanded and major improvements have been made to the efficiency of many key algorithms.

  • Testing for isomorphism and equivalence of pairs of polytopes is now supported.

  • The automorphism group of a polyhedron may now be computed.

  • The f-vector of any polyhedron, along with the set of all i-faces and the face graph may now be computed.

  • The introduction of an improved algorithm has led to a big reduction in the time taken to compute the Hilbert basis of a cone.

  • Possible Minkowski decompositions of a lattice polygon may now be calculated.

  • Databases containing the classification of smooth Fano polytopes up to dimension 8, and canonical Fano polytopes in dimension 3, are included.

Group Theory

• Matrix Groups (Over Finite Fields)

  • The Composition Tree (CT) package developed by Henrik Bäärnhielm, Derek Holt, Charles Leedham-Green and Eamonn O'Brien, working with numerous collaborators, is being released for the first time. This package is designed for computing structural information for large degree matrix groups potentially having very large orders. The traditional approach involves constructing a base and strong generating set where the orbits of the base points have to have length at most a million and so is limited to small groups. The CT package uses a constructive version of Aschbacher's Theorem to construct a composition series for the group represented in the form of a tree.

  • Code built on top of the CT package allows the user to determine basic properties of the group such as the derived subgroup, chief series, and Sylow subgroup.

• Matrix Groups (Over Rings of Characteristic 0)

  • Code has been developed to compute the normaliser or centraliser of a finite subgroup of GL(n,Z).

  • Machinery is provided which enumerates the isomorphism classes of ZG-lattices, in the case in which the endomorphism ring of the group G is a number field.

  • A new package, "Infinite", has been developed by Alla Detinko, Dane Flannery and Eamonn O'Brien for groups defined over number fields, or (rational) function fields in zero or positive characteristic. This package contains algorithms to decide finiteness; if the group is found to be finite, an isomorphic matrix group defined over a finite field is returned. It also contains algorithms to decide nilpotency, solubility, the Tits Alternative and other "virtual" properties for such groups.

  • It is now possible to test irreducibility and primitivity of large finite nilpotent matrix groups defined over a number field or a rational function field. The method does not use either the BSGS or CT representations and is due to Tobias Rossmann.

• Permutation Groups

  • An implementation of the "Jellyfish" algorithm of Law, Niemeyer, Praeger, and Seress was written by Bill Unger. This will form part of a composition tree approach to permutation groups in Magma. The Magma implementation extends the original algorithm by providing a tolerably efficient inverse image facility.

• Classical Groups

  • Quadratic spaces have been implemented.

  • Clifford algebras have been implemented as structure constant algebras, using the above quadratic space machinery. An immediate application will be to the construction of the spinor group associated with certain orthogonal groups.

  • Witt's theorem has been implemented for orthogonal and symplectic geometries over finite fields, including fields of characteristic two. That is, an isometry defined on a subspace can be extended to an isometry of the entire space.

  • A standard generating set can be constructed for any classical group given in its natural representation.

  • A presentation for a classical group can be constructed on its standard generators.

Lattices and Quadratic Forms

• Lattices

  • A new version of the backtrack search algorithm for computing the automorphism group of an integral lattice has been developed. This can handle lattices having much larger sets of vectors of minimal norm than the Souvignier version which has been widely used up until the present.

  • Functionality is provided to compute the Hermitian and quaternionic automorphism groups of lattices when applicable (though the functions themselves work on the Gram matrix of such lattices).

• Quadratic Forms

  • Code has also been developed for computing automorphism groups (testing isometry) of definite quadratic/bilinear forms over F_q[t], with q odd.

L-functions

• Recent work by Tim and Vladmir Dokchitser enables Magma to work with Artin representations for much larger groups. It is now possible to apply the machinery to groups of order 50000.

• The above improvement makes computations with Dedekind ζ-functions much more efficient than was previously the case.

• Improvements have been made to the functions for working with L-functions of twists of elliptic curves by Artin representations. This makes the calculation of analytic rank and related objects much more efficient.

• The p-adic L-function of an elliptic curve at a given prime can now be computed.

Linear Algebra

• Matrices over Algebras

  • Basic operations are now supported for matrices defined over a quaternion algebra.

• Sparse Matrices

  • The operations on sparse matrices have been greatly expanded to include many more of the standard operations provided for dense matrices.

Lie Theory

• Reflection Groups

  • The package for complex reflection groups has been rewritten to ensure that the matrix entries belong to the ring of definition.

  • A complex root datum has been implemented for every finite complex reflection group. This is a generalisation of the "Demazure" root datum.

• Coxeter Groups

  • A highly efficient algorithm for computing the growth function of a Coxeter group was developed and implemented by Bob Howlett and Bill Unger.

• Lie Algebras

  • Code has been developed to compute a Chevalley basis for a Lie algebra in characteristics 2 and 3. Taken together with previous work, the construction of a Chevalley basis is now available in every characteristic.

  • An experimental algorithm for calculating split toral subalgebras in characteristics 2 and 3 has been released.

  • A construction for Melikyan Lie algebras has been implemented.

Numerical Analysis

• Eigenvalues of a matrix may now be computed to a fixed precision using a stable numerical algorithm.

• Numerical approximations of eigenvectors of a given approximate eigenvalue can be computed in cases that are not ill-conditioned

• The numerical derivative of a function can be computed in cases where this is numerically workable.

Representation Theory

• A package has been developed which constructs the principal indecomposable modules for a group algebra over a given finite field.

• The Meataxe-based algorithms now detect any diagonal block structure in the input and use the corresponding decomposition to speed up matrix multiplication.

System

• Runtime System

  • The memory manager for Magma has been upgraded significantly by the introduction of a new Mark-Sweep garbage collection scheme. This leads in general to less memory usage for many typical computations, especially those involving long loops over databases. In particular:

    • Leaks of objects arising from circular references from attributes defined in package code have been removed.

    • Leaks of objects arising from internal circular references have been removed.

    • Slowdowns involving algebraic number fields (caused by complex circular reference analysis) have been removed.

  • A large number of leaks in internal algorithms have also been fixed (independent of the above garbage collection scheme).

• Language

  • The extended types machinery now supports arbitrary levels of nesting for sets, indexed sets, and multisets. (Formerly only sequences would nest.)

4.1 Schemes

New Features:

• Multiplicities for singular points on arbitrary schemes over a field are now available. The computation uses local Groebner bases.

• Tangent Cones are now computed for general schemes (the old version was really only guaranteed correct for hypersurfaces). This again makes use of the new local Groebner basis functionality.

Changes and Removals:

• Affine patches of subschemes of length 1 schemes are not available (as has always been with subschemes of length 0 schemes).

• The PointSearch function had a few bug fixes and some improvements.

4.2 Sheaves

New Features:

• Functions SheafOfDifferentials and TangentSheaf have been added to compute the sheaf of 1-differentials and the tangent sheaf of ordinary projective varieties. Along with other applications (computation of the dimension of the connected component of automorphism groups or the dimension of deformation spaces), combined with the IsLocallyFree intrinsic, this gives an alternative method to test for non-singularity that can be superior to the default Jacobian method for low dimensional varieties in high dimensional projective spaces.

4.3 Algebraic Surfaces

New Features:

• Functionality has been added to handle the parametrization of degree 3 and 4 Del Pezzo surfaces in a more efficient manner. This also fixes some incorrect processing in the old version that resulted in some singular parametrizable surfaces being labelled as non-parametrizable. The new methods work more directly and generally avoid blowing down exceptional curves to give higher degree surfaces. This is incorporated in the general Del Pezzo parametrization routine, but can also be accessed through intrinsics ParametrizeSingularDegree3(4)DelPezzo.

4.4 Algebraic Curves

New Features:

• MatrixRepresentation is a new function that gives the representation of a curve automorphism group on the space of global differentials when the genus is at least 2.

• Maps on curves: a scheme map from a curve with function field into a general scheme will now automatically try to compute the image of points in the base scheme of the map using the function field place machinery. Basically, if the map on the projective normalisation of C is defined at all points over the given point p (always true if the codomain is projective) and these points have the same image, which is also defined over the base field of the point set of p, then that will be returned as the image of p. This will give the correct result at all points to which the map can be extended, but avoids the expensive global Extend function.

• Tangent lines and cones, multiplicities of singular points and most of the other local singularity functions have been extended from plane curves to general curves.

• The intrinsic IsHypersurfaceDivisor returns whether an effective divisor on an ordinary projective curve is the scheme-theoretic intersection of the curve with a hypersurface. If this is the case, such a hypersurface is also returned.

4.5 Toric Varieties

New Features:

• The existing subscheme intrinsics Dimension, IsProjective, IsAffine, and EmptySubscheme, along with all intrinsics which build on them, give correct answers for a wider class of ambient spaces via toric computations. Although slower, this allows the user to work with, for example, subschemes in non-Q-factorial toric varieties.

• A well-known limitation with the Magma schemes is the strict definition of affine patch which makes working with graded spaces such as P(2,3,5,7) problematic. In the toric case, several new intrinsics are provided to help. The intrinsic ToricAffinePatch will return the affine patch realised as a toric variety, along with the embedding. Similarly, intrinsics ToricFunctionField and ToricLiftRationalFunction do not require the traditional affine patch in order to perform computations.

• The BigTorus of a toric variety X can be recovered as a toric variety, along with its embedding into X and the rational map from X to the torus. The map of the big tori of the domain and codomain of a toric map can be recovered via UnderlyingToriMap.

• The intrinsic RestrictionToSubtorus(X) returns the restriction of a subscheme Z to the largest subtorus of the ambient containing Z.

• A toric map can be tested for being the identity map via IsIdentity.

• The toric ideal generated by a cone C can be created using the Ideal intrinsic.

• The non-Q-factorial locus of a toric variety can be obtained from NonQFactorialLocus.

• The computation of the (geometric) genus of a curve realised as a subscheme in a toric ambient space can be computed using the toric (rather than scheme) methods via GeometricGenusUsingToricGeometry.

Changes and Removals:

• Intrinsics IsTerminal, IsCanonical, and IsGorenstein are now substantially faster in the (fake) weighted projective space case, through use of the inequalities of arXiv:0805.1008 to minimise the amount of singularity analysis.

• An improvement to the convex hull algorithms means that computing resolutions of singular toric varieties is much faster. The algorithm is deterministic by default, with an optional parameter to make use of randomisation.

• The intrinsics IsWeightedProjectiveSpace and IsFakeWeightedProjectiveSpace no longer compute the fan; the result is read from the Cox data.

• The intrinsic DirectProduct now avoids creating the fan. If the fan is required later, it will be constructed using the fans of the product varieties. Because of the closer integration with the toric calculations it is possible to compute the direct product of a much wider class of ambient schemes (where the scheme is in fact a toric variety).

• The parameters relevant and homogeneous have been added to map constructors.

5.1 Elliptic Curves

5.2 Genus One Models

New Features:

• Local solubility testing: IsLocallySolvable is implemented using efficient special-purpose code for models of degree 2,3 and 4. If the option Random is set to true, the local points found are randomised, as needed in descent-related routines.

• For genus one curves of degree higher than 4, MinimizationMatrix finds a change of coordinates to a minimal or near-minimal model. (Not guaranteed to work in all situations.) This was developed for use in the 6-, 8- and 12-descent code.

• Models of degrees 2,3 and 4 can be 'added" (in the sense of addition in the Selmer group) using AddModels.

• Intrinsics SL4Covariants, RemoveCrossTerms, AddCrossTerms, TransformationtoMap, CoveringMap are included.

Changes:

• Transformations between genus one models now have their own type, TransG1. All relevent intrinsics now accept/return this type.

• CompleteTheSquare has changed slightly, and returns a transformation.

5.3 Hyperelliptic Curves

New Features:

• L-series can now be (automatically) computed for most hyperelliptic curves.

Bug fixes:

• A bug in constructing a map from a hyperelliptic curve to its ambient has been fixed.

5.4 L-Series

New Features:

• The L-series of a Hilbert modular form can now be constructed.

• Automatic construction of the L-series of a hyperelliptic curve defined over Q is now possible. This uses the regular model machinery, and works in considerable generality.

Bug fixes:

• Some fixes and speedups for symmetric powers particularly in the case of modular forms have been made.

• Faster code from Anton Mellit has been added for symmetric powers and tensor products.

• A typo with EulerFactor for a modular form has been corrected.

• A bug with eq that could cause an infinite loop was fixed.

6.1 Modular Forms

Changes and Removals:

• Functions such as DeleteAllAssociatedData and DisownChildren for spaces of modular forms and modular symbols are deprecated. They are no longer needed, since effective garbage collection is now provided by the system which, in particular, handles circular references which previously prevented such objects from being deleted.

6.2 Modular Symbols

Changes:

• The intrinsic NewSubspace has been improved through the use of better linear algebra techniques.

• A lot of unnecessary work has been avoided in NewformDecomposition.

• The routine ModularSymbols(E) for an elliptic curve E has also been improved through the use of better linear algebra techniques.

Bug fixes:

• A bug in FullPrimaryComponents has been fixed, where an infinite loop occurred due to a failure to update the random seed.

6.3 Hilbert Modular Forms

New Features:

• Access intrinsics CentralCharacter and IsParallelWeight have been added.

• A Hecke-invariant InnerProduct is implemented for some spaces.

• A HeckeEigenvalueBound is implemented for cuspidal newforms.

• The LSeries of a cuspidal newform is implemented.

• New or improved algorithms have been implemented in the following areas. Improvements in the precomputation for the "definite" method are indicated above. The non-precomputation part of it has been implemented more efficiently; in particular, parallel weight 2 is handled by separate code and is reasonably efficient. The inner product is used in several places to save work; Hecke eigenvalues can be computed without computing all columns of the corresponding Hecke operator on the full space. Better techniques for dealing with abstract Hecke modules over number fields have been implemented; work on this is ongoing.

• An option is provided to suppress taking the quotient by the Eisenstein series (which arise naturally in the "definite" algorithm for weight 2). It is invoked by setting RemoveEisenstein:=false when creating the space. This makes computations faster for large spaces.

• DeleteHeckePrecomputation is provided, to allow the user to conserve memory when necessary by deleting bulky "precomputed" data after it has been used. (This data is used to compute Hecke operators at given primes, for all spaces over the same number field. The precomputation phase to produce this data is the most expensive step in computing each space.)

Changes and Removals:

• More signatures for creating HilbertCuspForms are provided, for convenience.

• The behaviour of SetRationalBasis has changed: in particular, the circumstances in which this happens automatically have changed. (This feature may become completely automatic in a future version.)

• The method used to compute a NewSubspace can now be controlled by the user, via the Al option. The options are Degeneracy (degeneracy maps are used), or Naive (the old space is recognised as a Hecke module and split off).

Bug Fixes:

• Some runtime errors that occurred during the precomputation phase of the "definite" algorithm for fields of large discriminant have been fixed.

7.1 Algebraic Number Fields

New Features:

• The GaloisGroup of a number field defined over a number field which is itself a relative extension can now be directly computed.

• The GaloisGroup of a reducible polynomial over a number field can be computed. It can now also be computed over any extension of a number field.

Changes and Removals:

• A number field of degree 1 can be obtained using RationalsAsNumberField(). However,
  NumberField(Rationals()) now returns Rationals(). (This change is necessary due to compatibility issues.)

• The computation of Subfields of a number field represented as an extension of another number field has been replaced by a new algorithm being developed by Klüners and van Höij. Subfields can now be computed of any number field defined by a single polynomial.

• The (optional) use of the older algorithm to compute Galois groups has been restricted (mostly) to the GaloisGroup signatures which take a field as input rather than a polynomial.

• The algorithm used to compute maximal orders of Kummer extensions has been extended to also compute maximal orders of radical extensions.

• The computation of Dedekind ζ-functions has been improved through the use of Artin representations.

Bug Fixes:

• The computation of residue fields in orders having a multiply relative representation has been fixed

• A failure in the evaluation of a polynomial in the straight-line representation for elements not in the coefficient ring has been fixed.

• A fix has been made to addition of orders.

• Some fixes have been made to the computation of Galois groups.

7.2 Algebraic Function Fields

New Features:

• Subfields can be computed for all global function fields which are defined by a single polynomial.

• GaloisGroups of function fields which are extensions of global function fields by a single polynomial can now be computed.

• The intrinsics Completion and Expand are now available for function fields represented as a relative extension and elements thereof.

Changes and Removals:

• The GaloisGroup and Subfields computations for global function fields have been improved.

• The (optional) use of the older algorithm to compute Galois groups has been restricted to the GaloisGroup signatures which take a polynomial as input rather than a field.

• The algorithm used to compute maximal orders of Kummer extensions has been extended to also compute maximal orders of radical extensions.

• It is now possible to compute the maximal order of an Artin-Schreier extension which is an extension of an extension of a rational function field.

• Orders of function fields are now recognized as being domains and some calculations such as matrix determinants over these orders will speed up as a result.

Bug Fixes:

• A fix has been made to GaloisGroup for characteristic p fields.

• The bug in the calculation of the genus of a function field in non-simple representation has been fixed.

• A bug in the computation of residue fields of orders having a multiply relative representation has been fixed.

• A failure in the evaluation of a polynomial in the straight-line representation in the case of elements not in the coefficient ring has been fixed.

8.1 p-adic Rings and their Extensions

Bug Fixes:

• A bug in IsDivisibleBy for local ring elements has been fixed.

• The Log intrinsic has been extended to arbitrary units rather than 1-units only.

8.2 Series Rings

Changes and Removals:

• A fix in the precision handling of field elements has led to some field elements having a reduced (but correct) precision. One effect of this is that polynomials over a series extension field may no longer have enough precision to be used to construct an extension (where full precision in each coefficient is required). This problem can be avoided by putting the coefficients of the polynomial into a sequence and calling Polynomial on this sequence. This avoids the addition of any O terms to individual terms of the polynomial with the consequent loss of precision. (x is really x + O(t^field_prec), where x is a polynomial variable and t is the generator of the series field or extension).

Bug Fixes:

• Factorization of inseparable polynomials over series rings has been fixed.

• A bug in giving a series the full precision of the ring has been fixed.

• Fixed a bug which could causes Evaluate to return an answer with too much precision when evaluating a series at another series.

8.3 Lazy Series Rings

Bug Fixes:

• A bug in the computation of Coefficients when the range is empty has been fixed.

8.4 General Local Fields

New Features:

• The intrinsic RelativePrecision has been added for elements of general local fields.

• The intrinsic Degree taking a general local field and a coefficient ring of that field has been added.

• The intrinsic RelativeField has been added which computes a local field as an extension of a subfield given a field and a map from the subfield into the local field.

Changes and Removals:

• The mapping to the ramified representation of a general local field has been improved.

• The intrinsic FixedField of a general local field has been reimplemented using invariants. Precision issues in FixedField are now handled better.

• Loss of precision in some operations in General Local Fields has been reduced, specifically in application of automorphisms, although as a result they are no longer created as hom<> objects.

Bug Fixes:

• Fixes to the RamifiedRepresentation map have been made for fields of degree 1.

• The mapping of a general local field to its ramified representation has been fixed to avoid unnecessary precision loss.

8.5 Algebraic Power Series

New Features:

• Arithmetic operations may now be specified using the infix symbols +, - and *. Equality testing can be specified using the operator eq.

Changes and Removals:

• The parent of a RngPowAlgElt is now a power structure and the type RngPowAlg is no longer used. The type RngPowAlgElt has been changed to SerPowAlg.

9.1 Integer Ring

Changes and Removals:

• The computation of Binomial is now permitted when both arguments are large but the result is still relatively small.

• The behaviour of ShiftRight on negative integers has been adjusted to make it consistent across all shift ranges.

9.2 Finite Fields

New Features:

• The algorithm for factoring a polynomial over a finite field when all the coefficients lie in a proper subfield has been improved.

• The finite field relationship machinery has been significantly modified so that fewer auxiliary fields are retained in the background and the updating of relationship information has been sped up.

• One may now use sub to construct a subfield of a finite field K where S is a sequence or set of elements coercible into K.

9.3 Real and Complex Fields

New Features:

• The computation of the NumericalDerivative of a function is now possible for well-behaved functions.

• The NumericalEigenvectors corresponding to an approximation of a complex eigenvalue can now be computed in cases that are not ill-conditioned.

• HypergeometricSeries2F1 now allows rational and integral values of the complex argument.

Changes and Removals:

• Complex arithmetic in Magma has been updated to use MPC 0.8.

• The inverse trigonometric functions and power functions have been updated to use MPC 0.8.

9.4 Polynomial Rings

New Features:

• For various fundamental algorithms (such as GCD and factorization), special efficient techniques are now used when it is detected that the input involves symmetric polynomials.

• Multivariate polynomial GCD has been sped up in the case when the degree in one variable is much larger than that in the other variables.

• A p-adic method is now used for powering multivariate polynomials over finite fields (and other inefficiencies have been fixed).

10.1 Linear Codes over Finite Fields

Changes and Removals:

• The speed of the MacWilliams transform has been improved in some cases.

• The computation of the weight enumerator or the number of words of specific weight now uses the dual code information more effectively in several cases.

Bug Fixes:

• Some memory leaks in the creation of QR codes were fixed.

• ConstacyclicCode now raises an error if a shift factor of zero is specified.

• Fixed a crash in the verbose output of CanteautChabaudsAttack.

10.2 Linear Codes over Finite Rings

Changes and Removals:

• The intrinsic KernelZ2CodeZ4 has been improved.

• The Zimmermann option for the minimum weight of a ℤ₄ code has been removed.

Bug Fixes:

• The intrinsics LeeWeightEnumerator and EuclideanWeightEnumerator now raise an error instead of crashing if the code is not defined over ℤ₄.

10.3 Additive Codes

New Features:

• The two intrinsics IsProjective and IsAdditiveProjective have been added to handle the two possibilities for additive codes.

Bug Fixes:

• Fixed a bug in the computation of meet when an additive code is involved.

• Fixed a bug in the computation of CodeComplement when using an additive and a linear code.

10.4 Quantum Error-Correcting Codes

Bug Fixes:

• The dimension of quantum codes is now returned as a rational.

• Fixed a bug in the computation of the minimum weight when using the Zimmermann algorithm.

• Fixed the verbose printing of which words were impurities in the minimum weight computation.

11.1 Symmetric Functions

Bug Fixes:

• Several crashes or erroneous successes when coercing into a symmetric function algebra have been fixed.

11.2 Hadamard Matrices

New Features:

• The database of Hadamard matrices has been expanded. New matrices have been added for degrees 36 (1), 48 (60) and 60 (1759). The additional matrices were supplied by Dragomir Djokovic.

11.3 Graphs

New Features:

• Intrinsics NumberOfVertices and NumberOfEdges added.

Bug Fixes:

• MinimumCut now returns an error instead of crashing when the source and sink vertices are the same.

12.1 Polynomial Rings

New Features:

• New function ElementToSequencePad (abbreviation EltseqPad) to return the coefficients of a univariate polynomial quotient ring, with zero padding.

Bug Fixes:

• Fixed a bug that could occasionally cause the constant term of a polynomial defined over the complex numbers to be printed incorrectly.

• Fixed a bug in the root computation of polynomials over local fields where the polynomial was not coercible into the corresponding local ring.

13.1 Toric Lattices

New Features:

• A map between toric lattices can be tested for being the identity map via IsIdentity.

• The value v·v for a lattice point v can be computed more efficiently using the new Norm intrinsic.

• Many operations on sequences of lattice points are now also possible on sets.

• A lattice point can now be multiplied on the right by a square matrix with rational or integer entries.

13.2 Fans

Changes and Removals:

• The intrinsic NormalFan has been added as an alias for the DualFan of a polytope.

• The intrinsic DualFaceInDualFan now accepts a face defined by a polyhedron, rather than simply a sequence of (ray) indices.

• The intrinsic NonSimplicialCones allows fast extraction of the non-Q-factorial cones from a fan.

• Improved speed on fan creation from Cox data.

Bug Fixes:

• Fixed a bug in the creator of the Cox data from a degenerate fan.

13.3 Cones

New Features:

• The intrinsic MatrixOfInequalities returns the inequalities of a cone arranged in a matrix, with each row representing an inequality.

• The intrinsic RandomPositiveCone can be used to create a random cone.

• The intrinsic NormalCone(P,F) has been added to return the (outer) normal cone to a face F of a polyhedron P.

• The intrinsic BoxElements generates the points in the fundamental domain of a simplicial cone.

• The toric ideal associated with a cone can be constructed via Ideal.

Changes and Removals:

• An improved version of Hemmeke's algorithm for calculating minimal generators of a semigroup has been implemented. This can be called directly via the MinimalPositiveGenerators intrinsic, and is used indirectly when computing the Hilbert basis of a cone.

• An improved version of the intrinsic ZGenerator makes intelligent use of Hemmecke's algorithm when computing the Hilbert basis.

• MinimalPositiveGenerators can now handle huge gradings.

• Point finding in a cone is now significantly faster and requires a great deal less memory.

• A cone can be multiplied on the right by a square matrix over ℤ or ℚ.

Bug Fixes:

• Removed memory leaks associated to the Barvinok algorithm.

• Given an identity map id and cone C, Image(Id, C) now returns C rather than constructing a copy of C.

• The ambient lattice of the data returned by QuotientGenerators has been fixed to be mathematically consistent with the underlying maps.

13.4 Polytopes and Polyhedra

New Features:

• A pair of polytopes can be tested for isomorphism or equivalence.

• The hVector for a simplicial polytope can now be computed.

• The intrinsic IsFace(P,F) is provided for polyhedra.

• Perfect centring for a polytope P may be determined using the intrinsic IsPerfectlyCentered.

• The Minkowski decompositions of a lattice polygon can be computed.

• The minimum lattice width of a polytope can be computed.

• Intrinsics Degree and Codegree are provided for lattice polytopes.

• Added IsFlag for a simplicial polytope.

• The intrinsic VertexFacetHeightMatrix returns the lattice height of each vertex of a polytope with respect to each facet. The VertexFacetIncidenceMatrix can be used to tell which vertex is contained in which facet.

• Several databases of polytope classifications have been included. For further details see the online Graded Ring Database (http://grdb.lboro.ac.uk/).

• A polytope can be saved in PALP format for use with Kreuzer and Skarke's polytope analysis software.

• The ViewWithJavaview intrinsic can be used to visualise a polytope (of dimension at most three) using the Javaview software, if installed.

Changes and Removals:

• The intrinsic EhrhartCoefficient now accepts negative dilation factors.

• The intrinsic IsIntegral has been modified to avoid calculating the vertices unless absolutely necessary, improving the speed of a number of operations.

• The intrinsics InteriorPoints, NumberOfInteriorPoints, and NumberOfBoundaryPoints now make better use of reflexivity when appropriate. Special-case low-dimensional algorithms improve point finding and counting in many cases.

• The intrinsic fVector now uses the results of arXiv:1002.2815v2 in the low-dimensional smooth cases.

• Ehrhart calculations for smooth Fano polytopes now take advantage of the results in
  arXiv:1004.3817v1.

• Ehrhart calculations for low-dimensional reflexive polytopes now take advantage of the results in arXiv:1002.2815v2.

• The time taken for computing the volume of a simplicial polytope has been improved. More generally, the algorithm may choose to make use of the Ehrhart data.

• The volume of a 0-dimensional polytope is now defined to be 1 rather than 0.

• A polyhedron can be multiplied on the right by a square matrix over ℤ or ℚ.

Bug Fixes:

• The algorithm for computing the volume of a one-dimensional polytope (i.e. line segment) didn't handle the case when the end points were not integral. This has been corrected.

14.1 Finite Groups

• Code has been developed by Derek Holt and Bill Unger for constructing a semidirect product of two finite groups.

• A utility function for DicyclicGroup has been added.

Changes and Removals:

• The two functions G/N and quo have been made equivalent, so the first now returns the quotient epimorphism, as the second has always done. In the case of quotients of permutation and matrix groups, where the return type is always a permutation group, that group need not be a regular group as previously has been the case.

• The memory behaviour of the straight-line program representation of group elements has been revised, thereby greatly improving the speed when there are many elements in use.

• The printing format for a subgroup lattice has been inverted. The full group is now printed on the first line, and the trivial subgroup is printed on the last line. The numbering of the subgroup classes has been left unchanged.

Bug Fixes:

• A number of errors in the filtering stage of the Subgroups intrinsic have been fixed, particularly in the cases of the IsNilpotent, OrderEqual and OrderDividing filters.

14.2 Permutation Groups

New Features:

• An implementation of the Law-Niemeyer-Praeger-Seress "jellyfish" algorithm is available. The initial construction is by the JellyfishConstruction function. If this function succeeds, the JellyfishImage function maps from the large degree group to the small, while JellyfishPreimage provides an inverse.

14.3 Matrix Groups – General

New Features:

• For matrix groups represented in terms of a base and strong generating set, the Representation function now returns this data structure. In a similar vein, the WordInStrongGenerators function has been extended to matrix groups.

• For matrix groups represented in terms of a base and strong generating set, Magma makes more effort to retain a useful base when constructing the image and kernel of homomorphisms and when performing ChangeRing operations.

• Matrix groups over quaternion algebras can now be constructed, and various computations can be performed. These include: conjugacy testing and character tables, invariant forms, isomorphism testing, and G-modules.

14.4 Matrix Groups Over Finite Fields

The Composition Tree (CT) package developed by Henrik Bäärnhielm, Derek Holt, Charles Leedham-Green and Eamonn O'Brien, working with numerous collaborators, is being released for the first time. This implements the Composition Tree representation for a matrix group defined over F_q. It is an alternative representation when it is not practical to compute a BSGS for a matrix group defined over F_q.

New Features:

• The intrinsic CompositionTree and its associated functions allow the user to construct and manipulate a composition tree data structure for a matrix group defined over F_q.

• A number of functions have been implemnted by Derek Holt that take as their starting point the CT representation and construct the "trivial Fitting" model (TF model) for a matrix group defined over F_q.

• Holt has further developed a first set of functions for working with this TF model. For example, the intrinsic LMGCompositionSeries constructs a composition series for the group. Similar intrinsics have been provided which compute such things as a chief series, soluble radical, the derived subgroup, the centre and the Sylow p-subgroups.

• A number of functions for computing orders of standard matrix groups over finite fields have been modified to work with factored integers. This can improve performance.

14.5 Matrix Groups Over Characteristic 0 Fields

A new package, "Infinite", has been developed by Alla Detinko, Dane Flannery and Eamonn O'Brien for groups defined over number fields, or (rational) function fields in zero or positive characteristic.

New Features:

• The intrinsic CongruenceImage constructs various types of congruence images – defined over a finite field – for a finitely-generated linear group defined over a number field, or a (rational) function field in zero or positive characteristic.

• The intrinsic IsFinite includes new algorithms to decide finiteness of a finitely-generated linear group G defined over a number field or a (rational) function field in zero or positive characteristic. If the group is finite, then IsIsomorphic constructs an isomorphic copy of G as a matrix group of the same degree defined over a finite field.

• The intrinsic HasFiniteOrder can now be used to test whether an invertible matrix defined over a (rational) function field in zero or positive characteristic has finite order.

• The intrinsics IsNilpotent, IsSoluble, IsCompletelyReducible and IsUnipotent can now be applied to groups defined over a number field, or a (rational) function field in zero or positive characteristic.

• Functions which decide "virtual" properties are available for the first time for a linear group defined over number fields and in some cases a (rational) function field in zero or positive characteristic. The functions are IsPolycyclicByFinite, IsSolubleByFinite, IsNilpotentByFinite, IsAbelianByFinite, and IsCentralByFinite.

• The intrinsics IsIrreducibleFiniteNilpotent and IsPrimitiveFiniteNilpotent decide irreduciblity and primitivity of finite nilpotent matrix groups defined over a number field or a rational function field.

• Given a finite subgroup of GL(n,ℚ), its GL(n,ℤ) conjugacy classes can be determined using the intrinsic ZClasses.

• Given two finite subgroups of GL(n,ℤ), the intrinsic IsGLZConjugate will determine whether they are conjugate. Similarly, IsGLQConjugate will determine whether two finite subgroups of GL(n,ℚ) are conjugate.

• A database of representatives of the GL(n,ℤ)-conjugacy classes of irreducible maximal finite subgroups of GL(n,ℤ) for n ≤ 11 and n∈{ 13,17,19,23 } is included.

Changes and Removals:

• The intrinsic EndomorphismRing now returns a subalgebra of ℤ^n×n when applied to subgroups of GL_n(ℤ).

14.6 Finite Chevalley Groups

Changes and Removals:

• The intrinsic ChevalleyGroup for twisted groups has been changed to ensure consistency with ATLAS notation. In particular, the group ³D₄(q) is returned by either ChevalleyGroup("3D",4,q) or by ChevalleyGroup("3D",4,GF(q^3)) and the group ²E₆(q) is returned by either
  ChevalleyGroup("2E",6,q) or by ChevalleyGroup("2E",6,GF(q^2)). This is consistent with the current Magma implementation for Chevalley groups of type ²A_n (unitary groups).

• The order and factored order of any finite Chevalley group is now efficiently computed without setting up the corresponding group.

14.7 Classical Groups

New Features:

• Quadratic spaces have been implemented.

• Clifford algebras have been implemented as structure constant algebras, using the above quadratic space machinery. An immediate application will be to the construction of the spinor group associated with certain orthogonal groups.

• Witt's theorem has been implemented for orthogonal and symplectic geometries over finite fields, including fields of characteristic two. That is, an isometry defined on a subspace can be extended to an isometry of the entire space.

• The intrinsic ClassicalStandardGenerators returns the Leedham-Green and O'Brien standard generators for a classical group of specified type.

• The intrinsic ClassicalConstructiveRecognition constructs the standard generators for a classical group G = < X > in its natural representation as straight line programs (SLPs) in X.

• The intrinsic ClassicalStandardPresentation sets up a "short" presentation for a classical group on its standard generators.

• The intrinsic InvolutionClassicalGroupEven constructs an involution in G = < X > , a classical group in natural representation and even characteristic, and returns both the involution and its SLP in X.

14.8 Finite Soluble Groups

New Features:

• Two new intrinsics, MinimalNormalSubgroups and Socle, have been provided for finite soluble groups given by pc-presentations.

Bug Fixes:

• An error which would sometimes lead to the erroneous calculation of the lower p-central series of a group has been fixed.

• An error that could occasionally result in the incorrect calculation of the p-class of a p-group, arising either from the above problem or from the use of pCoveringGroup with a group having nuclear rank 0, has been fixed.

14.9 Finitely Presented Groups

New Features:

• The intrinsic DicyclicGroup constructs a dicyclic group of order 4q given either an integer q or a cyclic group of order 2q and an appropriate automorphism.

Changes and Removals:

• Changes to the way in which chains of subgroups are managed may improve execution times in some calculations.

• The various SimpleQuotients intrinsics have been made consistent regarding the parameters they accept, and the version which took a tuple argument has been removed.

14.10 Finitely Presented Abelian Groups

Bug Fixes:

• A bug in abelian groups which led to crashes when the order (or prime divisors of the order) exceeded 2³⁰ has been fixed.

14.11 Automatic Groups and RWS Groups

New Features:

• The AutomaticGroup function has two new parameters. The Huge parameter turns on/off the -h option of the KBMAG program.

  The IgnoreHaltingFactor parameter allows the user to disable stopping construction by halting factor.

Bug Fixes:

• A crash that occurred occasionally when converting a RWS group to sequences has been fixed.

15.1 Lattices

New Features:

• Hermitian automorphism groups (including quaternionic structures) can now be computed in some cases, though not on lattices per se, but rather on Gram matrices.

• The LatticeDatabase includes about 10 more extremal even unimodular lattices. These include a recently discovered 72-dimensional lattice from Nebe called (SL2(25)xPSL2(7)):2, two 80-dimensional lattices from Stehlé and Watkins called SL2(79) and (SL2(41).circ.S3\ ).2, three 64-dimensional lattices including one from coding theory by Ozeki called T64 (the others are "generic" lattices G64 and H64), and a cyclotomic lattice of dimension 56 called B_{56,1}^4 from a 1995 paper of Batut, Quebbemann, and Scharlau.

• The facility to obtain a lattice from a linear code over Z/mZ has been added.

Bug Fixes:

• A memory leak in the shortest vector computation was fixed.

• The behaviour of th EnumerationCost function has been modified in the context of search-tree pruning.

• A bug in the computation of the inverse of the quotient map of a lattice by a sublattice has been fixed.

• A bug giving incorrect results in ClosestVectors for a lattice with non-standard basis and non-trivial inner product has been fixed.

16.1 Reflection Groups

New Features:

• The intrinsic ComplexRootDatum computes a root datum for a finite complex reflection group.

• The implementation of pseudo-reflections has been revised to ensure consistency with the Bourbaki definition. (Pseudo-reflections include projections and transvections.)

• New functions SymplecticTransvection and UnitaryTransvection have been implemented.

Changes and Removals:

• ComplexReflectionGroup and ImprimitiveReflectionGroup have been deprecated and will be removed in a future release. Their functionality has been superseded by a new implementation of ShephardTodd. The old implementation of ShephardTodd has been renamed ShephardToddOld.

• The intrinsic IsPseudoreflection has been renamed to IsPseudoReflection to reflect the changed functionality.

16.2 Lie Algebras

New Features:

• The intrinsic ChevalleyBasis is now able to compute Chevalley bases for Lie algebras of algebraic groups over fields of characteristic zero as well as over finite fields of non-zero characteristic. Furthermore, there is a new variant that takes a root datum with respect to which the Chevalley basis should be computed as one of its arguments. Finally, the Chevalley basis for Lie algebras constructed by LieAlgebra(R, k) for a root datum R and a field k now correctly returns a Chevalley basis with respect to the root datum R.

• The new intrinsic IsChevalleyBasis tests whether a particular basis is a Chevalley basis.

• The intrinsic SemisimpleType now works in many cases for Lie algebras over finite fields of characteristic 2 or 3.

• The intrinsic LieAlgebraOfDerivations is provided to construct the Lie algebra of derivations for a structure constant Lie algebra.

• The new intrinsic ReductiveType is provided to recognize a reductive Lie algebra. It is designed to work in all characteristics, and for both split and twisted Lie algebras. Moreover, upon successful recognition a proof in the form of a Chevalley basis or a twisted basis is returned.

• Several intrinsics dealing with (split) toral subalgebras have been added, notably SplitToralSubalgebra, SplitMaximalToralSubalgebra, and IsSplitToralSubalgebra.

• The new intrinsic MelikianLieAlgebra is provided to construct Melikian Lie algebras.

• The new intrinsic QuotientWithPullback is provided to construct the quotient L/I of a Lie algebra L by an ideal I, togther with the pullback map from I to L.

• Several improvements have been made in the interface for computations with finitely presented Lie algebras. For example, it is now possible to construct homomorphisms from finitely presented Lie algebras and to construct quotients of such Lie algebras by relations.

Changes and Removals:

• The intrinsic TwistedRootDatum now accepts a string as first argument (whereas previously it would only accept a root datum), and it now also accepts a permutation for the optional argument Twist. Furthermore, when TwistedRootDatum is used to construct an untwisted root datum it emits a warning.

• In addition to the existing method for creating twisted Lie algebras by a split root datum R and a permutation p (using LieAlgebra(R, k, p)) it is now possible to construct twisted Lie algebras by a twisted root datum S (using Lie Algebra(S, k)) or, if the split root datum R is irreducible and not of type D₄, simply by TwistedLieAlgebra(R, k).

• Construction of a Cartan-type Lie algebra L by means of the intrinsics WittLieAlgebra, (Conformal)SpecialLieAlgebra, (Conformal)HamiltonianLieAlgebra, and ContactLieAlgebra now also returns a map between a polynomial ring and L, to aid in the identification of the basis elements of L.

• The intrinsics AdjointRepresentation and StandardRepresentation now take an optional argument ComputePreImage, which is true by default.

• The number of intrinsics for constructing Lie algebras has been reduced: SemisimpleLieAlgebra, SemisimpleMatrixLieAlgebra, ReductiveLieAlgebra, ReductiveMatrixLieAlgebra, SimpleLieAlgebra, and SimpleMatrixLieAlgebra are now deprecated in favour of LieAlgebra and MatrixLieAlgebra. For the time being, the deprecated intrinsics remain available with the suffix -Old.

Bug Fixes:

• Some assertion failures that could arise when using universal enveloping algebras have been fixed.

• Fixed the quo constructor for finitely presented Lie algebras.

• The intrinsic CompositionSeries for structure constant algebras has been sped up.

• Several small bugs in the intrinsic TwistedRootDatum were fixed.

• The intrinsic IsSplittingCartanSubalgebra has been sped up.

16.3 Universal Enveloping Algebras

Bug Fixes:

• Fixed some incorrect internal assertions when using universal enveloping algebras.

16.4 Representations

Bug Fixes:

• Three omissions in RestrictionMatrix were fixed. Bug reported and solution provided by R. Zeier.

• A problem in the arithmetic with representation decompositions was fixed.

17.1 Matrices

New Features:

• New function IsMonomial to test whether a matrix is monomial.

• Matrices over associative algebras have various added functionality, such as EchelonForm.

• Matrix rings/spaces now allow coercion of sequences of sequences of elements.

• The intrinsic Cofactors has been added for cases where the positional signs of the minors are wanted.

• The function IsDiagonal now also returns a sequence giving the entries on the diagonal when the matrix is diagonal.

• New function/procedure RemoveZeroRows.

Changes and Removals:

• The intrinsic Minors now returns the minors in adjugate order and no longer accepts a Signed argument.

Bug Fixes:

• The universe is now set correctly when RowSequence returns an empty sequence.

17.2 Sparse Matrices

The operations on sparse matrices have been expanded to include many more which correspond to the operations provided for dense matrices.

New Features:

• Sparse matrices with many zero rows now have a more compact representation, which results in less memory usage and some speedups.

• Arithmetic operations such as negation, scalar multiplication, inverse and powering are now supported.

• New function Eltseq for a sparse matrix to give its entries as tuples.

• The predicates IsZero, IsOne, IsMinusOne, IsScalar, IsDiagonal, IsSymmetric, IsUpperTriangular, IsLowerTriangular are now supported.

• New functions/procedures Submatrix (ExtractBlock), SubmatrixRange (ExtractBlockRange), InsertBlock, RowSubmatrix, RowSubmatrixRange, ColumnSubmatrix, ColumnSubmatrixRange have been provided to extract or replace submatrices.

• New functions/procedures SwapRows, SwapColumns, ReverseRows, ReverseColumns have been provided for row and column operations.

• New functions/procedures AddRow, AddColumn, MultiplyRow, MultiplyColumn.

• New functions/procedures RemoveRow, RemoveColumn, RemoveRowColumn, RemoveZeroRows.

18.1 Quaternion Algebras

New Features:

• Most features for quaternion algebras over number fields are now supported for rational function fields over finite fields of odd characteristic.

• Matrices and matrix groups over quaternion algebras (and general associative algebras in some cases) are now supported to some extent.

• The algorithm for computing right ideal classes now incorporates a range of new techniques which cut down the search. These include a refined mass formula, smarter search strategies, use of automorphisms, use of theta series, and some implementation speedups.

Changes and Removals:

• The intrinsic IsSplittingField(AlgQuat, FldQuad) has been removed. Instead IsSplittingField(Fld, AlgQuat) should be used.

• The comparison cmpeq is now available for quaternion orders as well as associative orders.

Bug fixes:

• A number of bug fixes have been made in code that determines the equality of orders for quaternion algebras over ℤ and polynomial rings.

• A bug in orders of quaternion algebras over rings other than ℤ has been fixed.

• Determining equality of ideals of orders of quaternion algebras over rings other than orders of number fields has been fixed.

• For ideals of orders of quaternion algebras, a bug in the intrinsic Conjugate has been fixed.

18.2 Matrix Algebras

New Features:

• It is now possible to compute the split basic algebra corresponding to a matrix algebra defined over F_q. This uses the Carlson-Matthews algorithm for computing a presentation for a matrix algebra.

Changes and Removals:

• The implementation of the Carlson-Matthews algorithm for computing a presentation for a matrix algebra defined over a finite field has been improved in a number of ways. This resulted in a substantial reduction in the runtime for harder examples.

19.1 K[G]-Modules

New Features:

• The Meataxe tools in Magma have been revised so as to take advantage of situations where the action matrices have a block diagonal structure. This can result in significant speedups.

• A package has been developed to compute the projective indecomposable KG-modules that project onto specified irreducible KG-modules, for a finite field K. Thus, the intrinsic ProjectiveIndecomposableModules(G, K) finds the projective indecomposables corresponding to the inequivalent irreducible KG-modules for G.

• The intrinsic ProjectiveIndecomposableDimensions(G, K) returns the dimension of each projective indecomposable KG-module.

• The intrinsic CartanMatrix(G, K) returns the Cartan matrix corresponding to the irreducible KG-modules. Similarly, AbsoluteCartanMatrix(G, K) returns the Cartan matrix corresponding to the absolutely irreducible KG-modules.

• The intrinsic DecompositionMatrix(G, K) specifies the absolutely irreducible constituents for the mod-p restriction of each ordinary irreducible character.

• The intrinsic BasicAlgebraFG constructs the basic algebra of a modular group algebra. The method has been successfully applied to groups having order up to several million.

• As a variant on the above feature, the intrinsic BasicAlgebraOfBlockAlgebra computes the basic algebra associated with any p-block of a modular group algebra.

19.2 Modules over Algebras

New Features:

• The function QuaternionicGModule allows the user to write a G-module over a quaternion algebra when a splitting is known.

19.3 Cohomology

New Features:

• Higher cohomology groups can be computed using the new intrinsic CohomologicalDimensions. This function determines the dimensions of Hⁿ(G,M), where M is a KG-module, using projective covers and dimension shifting.

• The intrinsic Ext computes Ext¹(M,N) for G-modules M and N.

• The availability of E = Ext¹(M,N) enables the user to construct module extensions described in terms of an element of E.