6.1 General Groups

Removals and Changes:

• The library gps100 has been withdrawn. It is subsumed in the much extended database SmallGroups (see below).

New features:

• A database of all groups having order at most 1000 (excepting orders 512 and 768) has been included in Magma. This database was prepared by Bettina Eick and Hans Ulrich Besche in 1997 using GAP, and incorporates directly the libraries of 2-groups of order dividing 256 and the 3-groups of order dividing 729 which were prepared by M.F. Newman and E.A. O'Brien. The functions available for accessing the database are

  • SmallGroupDatabaseLimit,

  • NumberOfSmallGroups(n),

  • SmallGroup(o,n),

  • SmallGroups(o), and

  • SmallGroupProcess(o).

  Functions IsEmpty, Current, CurrentLabel and Advance are provided for working with a small group process.

• The function CharacterDegrees employs Slattery's algorithm to determine the degrees of the irreducible characters of a p-group.

6.2 Finitely Presented Groups

New features:

• Given a finitely presented group G and normal subgroups H and K of G such that K ≤ H ≤ G, and a prime p, the function GModule(G,H,K,p) returns the K[G]-module M corresponding to the action of G on the largest elementary abelian factor of H/K of p-power order. If P is omitted, the module returned is the largest elementary abelian factor of H/K.

• Given a finitely presented group G and normal subgroup H of G and a prime p, the function Gmodule(G,H,p) returns the K[G]-module M corresponding to the action of G on the largest elementary abelian factor of H of p-power order.

6.3 Finitely Generated Abelian Groups

New features:

• Given finite abelian groups G and H, the function Hom(G,H) computes an abstract abelian group isomorphic to E = Hom(G,H). In addition, a transfer map is returned which converts an element of E to the actual homomorphism. Related functions are HomGenerators and AllHomomorphisms.

6.4 Finite Soluble Groups

New features:

• If the abelian groups G and H are given in terms of power-conjugate presentations, the function Hom(G,H) computes an abstract abelian group isomorphic to E = Hom(G,H). In addition, a transfer map is returned which converts an element of E to the actual homomorphism. Related functions are HomGenerators and AllHomomorphisms.

• The function CharacterDegrees employs Slattery's algorithm to determine the degrees of the irreducible characters of a p-group.

6.5 Permutation Groups

Removals and Changes:

• The library prmgps of primitive groups has been withdrawn. It is replaced by the database PrimitiveGroups (see below).

• The library trngps of transitive groups has been withdrawn. It is replaced by the database TransitiveGroups (see below).

New features:

• The various functions that compute conjugacy classes of subgroups of a permutation group G have been modified to compute the conjugacy classes of subgroups of a quotient group of G. Thus, if N is a normal subgroup of G, the function Subgroups(G,N) will compute a preimage in G of a representative for each conjugacy class of subgroups of the quotient group G/N. The optional parameters that apply to Subgroups(G) also apply to this variant.

• The primitive groups library prmgps has been converted into a database. Thus, the primitive groups of degree up to 50 are now directly accessible from within Magma (without having to load the library), and the names of the access functions have been redesigned. The new functions are:

  • PrimitiveGroupDatabaseLimit,

  • NumberOfPrimitiveGroups (replacing PrmNumberOfDegree),

  • PrimitiveGroup (replacing PrmGroup, PrmGroupSatisfying, PrmGroupOfDegreeSatisfying),

  • PrimitiveGroups (replacing PrmGroupsSatisfying, PrmGroupsOfDegreeSatisfying),

  • PrimitiveGroupDescription (replacing PrmInfo), and

  • PrimitiveGroupProcess (replacing PrmProcess, PrmProcessOfDegree).

  The functions IsEmpty, Current, CurrentLabel and Advance are provided for working with a primitive group process, replacing PrmProcessIsEmpty, PrmProcessGroup, PrmProcessLabel and PrmProcessNext.

• The transitive groups library trngps has been converted into a database. Thus, the Hulpke tables of transitive groups of degree up to 22 are now directly accessible from within Magma (without having to load a library), and the names of the access functions have been redesigned. The new functions are:

  • TransitiveGroupDatabaseLimit,

  • NumberOfTransitiveGroups (replacing TrnNumberOfDegree),

  • TransitiveGroup (replacing TrnGroup, TrnGroupSatisfying, TrnGroupOfDegreeSatisfying),

  • TransitiveGroups (replacing TrnGroupsSatisfying, TrnGroupsOfDegreeSatisfying), and

  • TransitiveGroupProcess (replacing TrnProcess, TrnProcessOfDegree).

  The functions IsEmpty, Current, CurrentLabel and Advance are provided for working with a transitive group process, replacing TrnProcessIsEmpty, TrnProcessGroup, TrnProcessLabel and TrnProcessNext. A new function TransitiveGroupName has been installed; it returns a string giving the name of the group in the Hulpke database.

• The function TransitiveGroupIdentification(G) returns the index in the transitive group database of the group that is isomorphic to a given transitive group G.

• A recursive algorithm due to Bill Kantor for computing the maximal normal p-subgroup of a permutation group is now available as the function pCoreKantor. Related functions are MEANS, ElementaryAbelianNormalSubgroup and pElementaryAbelianNormalSubgroup.

Bug fixes:

• A bad memory management technique for the partition-backtrack permutation group algorithms which caused very great loss of performance for large-memory runs has been fixed.

• A bug in the implementation of a new algorithm for computing normal subgroups of permutation groups (first installed in V2.3) has been fixed. The effect of this bug was sometimes to cause a crash and sometimes to result in some subgroups being missed. (Reported by Jurgen Klüners.)

• A bug that sometimes caused incorrect results to be returned when computing the conjugacy classes of elements of a soluble permutation group has been fixed. (Reported by Gregor Kemper.)

6.6 General Matrix Groups

New features:

• New function VectorSpace(G) to return the vector space on which the matrix group G acts naturally (assuming G is over a field; otherwise the same as the function RSpace).

• The function Stabilizer now works when given an element (vector) of a G-module and an appropriate matrix group.

• New function IntegralGroup(G) which returns a group of integer matrices conjugate to the rational matrix group G.

6.7 Matrix Groups over Finite Fields

The basic facilities provided by Magma for computing with matrix groups over finite fields depend upon being able to construct a chain of stabilizers. However, there are many examples of groups of moderately small degree where we cannot find a suitable chain.

An on-going international research project seeks to develop algorithms to explore the structure of such groups. The main theoretical underpinning of the project comes from the classification by Aschbacher (1984) of the (maximal) subgroups of GL(d,q) into nine families. Much of the research effort to date has been devoted to designing algorithms to decide whether G belongs to one of the eight families whose members have a normal subgroup preserving a "natural linear structure"; here, we plan to exploit this information to explore G further, ultimately producing a composition series for G. The final family consists of groups G which are almost simple modulo scalars; here the aim is to determine, a pair of efficiently computable inverse isomorphisms between G and a standard copy of the relevant simple group.

New features:

• A family of functions is provided for testing whether a group preserves a form modulo scalars. The main functions are ClassicalForms, SymmetricBilinearForm, QuadraticForm, and UnitaryForm. Corresponding functions are provided to return the scalars corresponding to the generators of the group of the form. These functions were implemented in Magma by Alice Niemeyer and Anthony Pye.

• A family of functions is provided for testing whether a group is a classical group in its natural representation. The main functions are RecognizeClassical, IsLinearGroup, IsSymplecticGroup, IsOrthogonalGroup and IsUnitaryGroup. These functions apply the Niemeyer-Prager classical group recognition algorithm as implemented in Magma by Alice Niemeyer and Anthony Pye.

• The function IsPrimitive determines whether a subgroup G of GL(d,q) acts imprimitively on the underlying vector space. The ancillary functions BlockSystem and BlocksImage return information about a proper block system and the action of G on such a block system, respectively. This group of functions was implemented in Magma by Eamonn O'Brien.

• The function IsSemiLinear tests whether a matrix group G acts as a semilinear group of automorphisms on some vector space. If G does act semilinearly, information about this action is provided by the functions DegreeOfFieldExtension, CentralisingMatrix, FrobeniusAutomorphisms, and WriteOverLargerField. This group of functions was implemented in Magma by Eamonn O'Brien.

• The function IsTensor tests whether a matrix group G preserves a non-trivial tensor product decomposition. If G does so, information about the decomposition may be obtained using the functions TensorBasis, TensorFactors, and IsProportional. This group of functions was implemented in Magma by Eamonn O'Brien.

• The function SearchForDecomposition searches for certain types of decompositions (corresponding to some of the Aschbacher families) of a matrix group with respect to the normal closure of a supplied subgroup. The following additional functions IsExtraSpecialNormaliser, ExtraSpecialParameters, ExtraSpecialGroup, IsSymmetricTensor, SymmetricTensorBasis, SymmetricTensorFactors, SymmetricTensorPermutations provide access to information about some of the decompositions. This group of functions was implemented in Magma by Eamonn O'Brien.

• The function IsOverSmallerField uses the Glasby-Howlett algorithm to decide if the absolutely irreducible group G ≤ GL(d,K) has an equivalent representation over a subfield of K.

• Given a group G of d×d matrices over a finite field E having degree e and a subfield F of E having degree f, the function WriteOverSmallerField returns a group generated by the matrices of G written as de/f×de/f matrices over F.

7.1 General Rings

New features:

• New functions GCD and LCM taking a set or sequence of general ring elements.

• The function IsDivisibleBy is now available for many ring element types for the first time.

• New function ExactQuotient for exact quotient (see IsDivisibleBy).

7.2 Integer Ring

The Schönhage-Strassen FFT-based algorithm for the multiplication of very large integers has been implemented. The crossover point (when this method beats the Karatsuba method) is currently 2¹⁵ bits (approx. 10000 decimal digits) on Sun Sparc workstations and 2¹⁷ bits (approx. 40000 decimal digits) on Digital Alpha workstations. The product of two arbitrary integers, each having one million decimal/ digits, may now be computed in 5.1 seconds on a 250MHz Sun Ultrasparc, or in 2.0 seconds on a 333MHz Digital Alpha workstation (a 64-bit machine).

Magma V2.4 also contains an implementation of an asymptotically-fast integer (and polynomial) division algorithm which reduces division to multiplication with a constant scale factor that is in the practical range. Thus division of integers and polynomials are now based on the Karatsuba and Schönhage-Strassen (FFT) methods when applicable. The crossover point for integer division (when the new method outperforms the classical method) is currently at the point of dividing a 2¹² bit (approx. 1200 decimal digit) integer by a 2¹¹ (approx. 600 decimal digit) integer on Sun Sparc workstations.

Changes:

• The function IsDivisible has been renamed to IsDivisibleBy.

• The function SquareFree has been renamed to Squarefree and SquareFreeFactorization has been renamed to SquarefreeFactorization; the old function names have been removed from the documentation but kept in the executable to allow old code to continue to work for this release.

• The function ChineseRemainderTheorem or CRT taking three integers has been removed to avoid confusion. (Solution still retains the original functionality.)

• The function ChineseRemainderTheorem or CRT taking three integer sequences has been changed so that it now only takes two integer sequences (the original "multipliers" sequence is now considered to consist of all ones). (Solution still retains the original functionality.)

Summary of new features:

• New Schönhage-Strassen FFT method for fast multiplication of very large integers.

• New asymptotically-fast division algorithm.

• The function IsSquare now works for any integer.

• New function Ilog to return the floor of the logarithm of an integer to a given integer base.

7.3 Finite Fields

Magma V2.4 contains a new packed representation for polynomials over GF(2) which is used to represent elements of finite fields of characteristic 2. This has led to dramatic speed-ups for computations within such fields. Magma V2.3 used a two-step representation where a large field was represented as an extension field of a Zech representation field, but this representation could only be used if the degree of the field had a suitable divisor and was not too large. The prime factorization of the degree of the field is irrelevant in the new representation so, for example, computation in fields of characteristic 2 with large prime degree is now very much faster.

A database of sparse irreducible polynomials over GF(2) has been constructed for all degrees up to 11000. Thus any finite field GF(2^k), for k within this range, may be created without any delay. The sparseness of the defining polynomial is utilized in the arithmetic. Computation in fields such as GF(2¹⁰⁰⁰⁰) and GF(2¹⁰⁰⁰⁷) (10007 is prime) is now quite feasible.

The Shoup algorithm for factoring polynomials over large finite fields has been implemented in V2.4. Combined with the great improvements to the finite field arithmetic itself, factorization over large finite fields is now very much faster in V2.4 than in previous versions. For example, factorizing a polynomial over GF(2¹⁰⁰⁰) is about 500–1000 [sic] times faster in V2.4 than in V2.3. For fields of higher degree, the improvement is even greater.

On a 64-bit 200MHz SGI Origin 2000, the n = 2048 polynomial from the von zur Gathen challenge benchmarks (of degree 2048 with sparse coefficients modulo a 2050-bit prime) is factored by Magma V2.4 in about 18.8 hours and the n = 3000 polynomial from the same benchmarks (of degree 3000 with sparse coefficients modulo a 3002-bit prime) is factored by Magma V2.4 in about 75.8 hours. Also, the n = 2048 polynomial from the Shoup challenge benchmarks (of degree 2048 with dense coefficients modulo a 2048-bit prime) is factored by Magma V2.4 in about 36.0 hours on the same machine. (The times taken for the same problems are about twice as long on a 250MHz Sun Ultrasparc.)

Summary of new features:

• New database of sparse irreducible polynomials over GF(2) for all degrees up to 11000 (accessed by IrreduciblePolynomial(GF(2), d) and also used internally by functions such as the creation function FiniteField).

• The arithmetic for univariate polynomials over finite fields, and arithmetic of medium-sized non-prime fields has been sped up by use of FFT methods, etc. (see Univariate Polynomial Rings below).

• Using a new efficient packed representation, arithmetic with finite fields of characteristic 2 has been dramatically improved.

• New Shoup factorization algorithm for polynomials over finite fields (automatically selected by Factorization when appropriate or by a parameter).

• The computation of square roots of elements of finite fields of characteristic 2 has been sped up.

• The function FiniteField(p,n) now has a parameter Check; setting Check to false will cause the function to skip the check that p is prime.

• New function NormEquation to solve norm equations in finite fields.

• New function AllRoots to return all n-th roots of a given element of a finite field.

• New function RootsInSplittingField(f), which, given a polynomial over a finite field K, computes a splitting field S of f as an extension field of K, and returns the roots of f in S, together with S.

• New function FactorizationOverSplittingField(f), which, given a polynomial over a finite field K, computes a splitting field S of f as an extension field of K, and returns the factorization (into linears) of f over S, together with S.

• New function Log(b, x) to compute the discrete logarithm of x to the base b, for any primitive element b.

• New procedure SetPrimitiveElement(K, x) to set the internal primitive element of K to be x (useful when computing many logarithms with respect to a particular base).

• The computation of discrete logarithms has been sped up (particularly for fields whose cardinality is less than 10¹⁰). The function Log may now be applied to an element of any finite field of arbitrary size. The Pohlig-Hellman algorithm for computing discrete algorithms is now combined with both the Shanks baby-step/giant-step and Pollard-ρ algorithms. In V2.4, computing the logarithm of any element of a field having cardinality less than 10¹⁰ takes less than a second, and computing the logarithm of any element of a field where the largest prime divisor of the order of the multiplicative group is 15 decimal digits now takes about 100 seconds (on a 250MHz Sun Ultrasparc).

7.4 Univariate Polynomial Rings

Magma V2.4 incorporates new fast methods for performing arithmetic with univariate polynomials. These include two FFT-based methods for multiplication: the Schönhage-Strassen FFT method where the coefficients are large compared with the degree, and the small-prime modular FFT with Chinese remaindering method where the coefficients are small compared with the degree. These new methods are applied to multiplication of polynomials over the integer ring, the rational field, integer residue class rings and prime finite fields. For some kinds of coefficients, at least one of the FFT methods beats the Karatsuba method as low as degree 32 or 64; for all of the kinds of coefficients listed, the FFT methods beat the Karatsuba method for degree 128 or greater.

An asymptotically-fast division algorithm (which reduces division to multiplication) is now used for polynomials over all coefficient rings (similar to the algorithm used for integers).

Factorization of univariate polynomials over all finite field types has been dramatically improved—see the section on Finite Fields above.

Changes:

• The function SquareFreeFactorization has been renamed to SquarefreeFactorization; the old function name has been removed from the documentation but kept in the executable to allow old code to continue to work for this release.

Summary of new features:

• New Schönhage-Strassen FFT algorithm for multiplication of polynomials (with large coefficients).

• New small-prime modular FFT algorithm for multiplication of polynomials (with small coefficients).

• New asymptotically-fast division algorithm.

• Modular exponentiation sped up by use of pre-inversion of modulus and fast multiplication (and thus leading to a speed up in all algorithms, such as factorization, which use this).

• New fast modular algorithm for the computation of the extended GCD of two univariate polynomials over the rational field.

• Factorization of polynomials over all finite fields greatly improved—see the section on Finite Fields.

• New parameter Global for polynomial ring creation functions. This now allows the creation of multiple non-global polynomial rings with different generator names.

• The function Evaluate for univariate polynomial rings has been changed and extended to use the common over-ring of its arguments; this overcomes some previous confusing behaviour caused by automatic coercion.

• The common over-structure of univariate polynomial rings is now supported for the first time (used by automatic coercion).

• New function HasPolynomialFactorization(R) to determine whether factorization of polynomials over the ring R is allowed in Magma.

• New function Valuation for univariate polynomials.

7.5 Multivariate Polynomial Rings

New algorithms for factorization of bivariate polynomials and general multivariate polynomials over finite fields have been implemented. The previous implementations failed on various inputs and were rather unsatisfactory.

Changes:

• The function Variety now returns a sequence of tuples.

• The function SquareFreeFactorization has been renamed to SquarefreeFactorization.

New features:

• The dimension of a full polynomial ring (considered as an ideal) is now defined to be -1, so the functions Dimension, IsZeroDimensional, etc. may be applied to such.

• New improved algorithm for factorization of multivariate polynomials over finite fields.

• New fast algorithm for the computation of GCDs of multivariate polynomials over all fields of characteristic zero using evaluation-interpolation techniques.

• New fast algorithm for factorization of multivariate polynomials over algebraic number fields (including quadratic and cyclotomic fields) by better use of resultants.

• New function TriangularDecomposition to compute the triangular decomposition of zero-dimensional ideals (algorithm of D. Lazard).

• The primary decomposition algorithm has been significantly improved by use of the triangular decomposition algorithm together with other techniques.

7.6 Invariant Rings of Finite Groups

New features:

• The functions IsInvariant(f, g) and IsInvariant(f, G) have been documented for the first time.

• The functions PrimaryAlgebra and PrimaryIdeal have been documented for the first time.

7.7 Algebraic Number Fields

The KANT number field machinery corresponding to Kash 1.9 has been installed. This is the first major upgrade of the Magma general number field facility since Kash V1.6 was installed in early 1996. This version of KANT provides the Magma user with greatly enhanced performance for many fundamental algorithms. Of particular note is the new KANT maximal order algorithm that combines both the Round 2 and Round 4 methods. Both conditional and unconditional computation of class groups has been greatly improved and the result is far superior performance to that provided in previous versions of Magma.

Major new number field facilities available in Magma V2.4 (courtesy of KANT) are ray class groups, S-unit groups, prime ideal decompositions of fractional ideals, automorphism groups of normal and abelian extensions, solution of Thue equations, unit equations and index form equations.

Changes:

• The function BetterPolynomial has been replaced by OptimizedRepresentation. The function BetterPolynomial will remain for a transition period.

• The function MaximalOrder has new parameters reflecting changes to the underlying algorithm.

• The function ClassGroup has new parameters reflecting changes to the underlying algorithm.

• The number field function BetterPolynomial has been replaced by OptimizedRepresentation which has slightly different semantics. The old function will be retained for compatibility in the medium term.

• All number field functions that return (inexact) real or complex numbers have been changed so that instead of returning fixed precision numbers, they now return free numbers (type FldPrElt).

New features:

• The function RadicalExtension allows easy construction of radical extensions.

• The function CompositeFields creates the composition of two number fields.

• The function SplittingField will construct the splitting field of a number field directly.

• The function RelativeField allows the construction of relative extensions.

• The function AbsoluteDegree gives the absolute degree of an order or number field.

• The functions IsNormal and IsAbelian test a number field for being a normal (respectively normal abelian) extension of ℚ.

• The functions CharacteristicPolynomial and AbsoluteCharacteristicPolynomial give the characteristic polynomial of a field element.

• The function Divisors, applied to an algebraic integer, returns all its divisors.

• The function Zeros gives the zeros of the defining polynomial of a field or order.

• The functions Index gives the index of an ideal in an over-order.

• The infix operator meet has been extended so as to compute the intersections of two ideals belonging to an order.

• The function Decomposition, applied to a rational prime p, finds the factorization of p into prime ideals in a designated order. If this function is applied to an order element a it finds the factorization of a into prime ideals.

• The function InertiaDegree gives the degree of inertia of a prime ideal.

• The function Verify is provided for checking a conditional class group computation.

• Unconditional and conditional (GRH) computation of ray class groups is provided by the function RayClassGroup.

• The function ClassRepresentative, applied to an ideal I belonging to an order whose class group is known, will give the chosen representative of the ideal class containing I.

• The group of S-units corresponding to a set S of prime ideals is provided by the function SUnitGroup.

• Function TorsionUnitGroup gives the torsion part of the unit group of a number field or order.

• The machinery developed by Klüners for computing subfields has been greatly improved. It has been sucessfully used to compute all subfields in an extension of degree 60.

• The functions Automorphisms and AutomorphismGroup provide for the calculation of the automorphism group of a normal extension.

• The machinery for solving Thue equations has been improved and an additional function ThueEval, for evaluating the homogeneous form, is provided.

Bug fixes:

• A number of bugs present in V2.3 and earlier versions, and arising in the computation of class groups, unit groups and testing ideals for being principal are fixed with the installation of the new version of KANT.

7.8 Rational Function Fields,

New features:

• The function Evaluate for univariate function fields has been changed and extended to use the common over-ring of its arguments; this avoids its previous confusing behaviour caused by automatic coercion.

• New function Derivative for function field elements (4 separate signatures).

• New function PartialFractionDecomposition which computes the complete partial fraction decomposition of a function field element. (PartialFractionDecomposition now factorizes fully and the original functionality is now given by the function SquarefreePartialFractionDecomposition.)

• Coercion from one rational function field to a compatible field is now allowed and the common over-structure of two function fields is computed correctly.

• New functions Degree, TotalDegree and WeightedDegree for function field elements.

7.9 Real and Complex Fields

New features:

• The function ExponentialIntegralE1 takes a real number r and calculates the principal value of

  ∈t_x ^∞eu⁄_udu
  at x = r.

Bug fixes:

• A bug in Xavier Gourdon's root finding program in the case of badly conditioned polynomials has been fixed.

7.10 Formal Series

The module for computing with formal series has been completely rewritten for Magma V2.4. Many bugs and leaks have been removed in the process.

The arithmetic for series over the rational field is generally 5 times faster (as a consequence of the use of fraction-free methods). Asymptotically-fast methods for arithmetic are now used for the first time (based on the new integer and polynomial methods). The 10000-th Bernoulli number B₁₀₀₀₀ may be computed in about 14 hours on a 250MHz Sun Ultrasparc directly from its generating function.

New features:

• Series with fractional exponents are now available. Arbitrary exponent denominators are allowed and such series may be freely mixed. Practically all series functions apply to series with fractional exponents if the result is meaningful and computable.

• The new category names for series rings and series are: RngSerPow (power series ring), RngSerPowElt (power series), RngSerLaur (Laurent series ring), RngSerLaurElt (Laurent series), RngSerPuis (Puiseux series ring), RngSerPuisElt (Puiseux series), RngSerPuis (any series ring), RngSerPuisElt (any series).

• The functions PowerSeriesAlgebra and LaurentSeriesAlgebra have been removed; the new creation functions are PowerSeriesRing, LaurentSeriesRing and PuiseuxSeriesRing.

• New parameter Global for series ring creation functions. This now allows the creation of multiple non-global series rings with different generator names.

• New function ExponentDenominator to return the denominator of exponents of a Puiseux series.

• New functions LeadingCoefficient and LeadingTerm.

• The function Reversion now works (for the first time) in the case of fields having non-zero characteristic and for series with general positive valuation.

• Intrinsics for the inverse trigonometric functions Arcsin, Arccos and Arctan and inverse hyperbolic functions Argsinh, Argcosh, Argtanh have been installed.

• Intrinsics for the trigonometric functions Sec, Cosec and Cot have been installed.

• The intrinsic function HypergeometricSeries is provided for computing hypergeometric series.

• New functions AGM, Gamma, LogGamma and Polylog.

• New intrinsic functions for computing with Elliptic and Modular functions have been implemented: Jacobi θ (JacobiTheta), Dedekind η (DedekindEta), j-invariant (jInvariant), Δ function (Delta), Eisenstein function (Eisenstein) and Weierstrass ℘-function (WeierstrassSeries).

8.1 Vector Spaces and R-spaces

New features:

• New function Moduli to return the column moduli of an R-space over a Euclidean domain.

8.2 R-Modules

New features:

• Function SingleMinimalSubmodule documented for the first time.

9.1 Group Algebras

Changes:

• The sub-constructor for group algebras now returns a sub-group algebra (in category AlgGrpSub) instead of an associative algebra. This is more natural.

• Ideals and subalgebras now make use of generators resulting in a very considerable speed-up in closure operations.

9.2 Matrix Algebras

New features:

• Genuine 64-bit versions for packed matrices over GF(2) have been implemented for 64-bit machines.

• The function Adjoint now works over any ring which has an exact division algorithm (and is now correct for the first time!).

• A matrix algebra element over a ring R is now coercible into a matrix group over another ring S if elements of R are coercible into S.

10.1 Elliptic Curves

The facilities for elliptic curves have undergone major expansion. Noteworthy is the introduction of general machinery for working with isomorphisms, isogenies and rational maps between curves. The implementation introduces new data types for such things as subgroups and subschemes of elliptic curves. The image of a subgroup under an isogeny (whose kernel is contained in the subgroup) can be calculated, an operation which is used to construct an isogeny from its kernel but is also of independent interest.

The implemented functions together form a powerful toolkit for working with elliptic curves. The interplay between elliptic curves, isogenies and subgroups is particularly beneficial. The toolkit allows comparatively easy computation of structures such as the endomorphism ring of a curve over a finite field and the eigenvalues of the Frobenius automorphism (the latter being useful in point-counting).

Changes:

• The category name for elliptic curves has been changed to CurveEll and the category name for elliptic curve points has been changed to CurveEllPt.

11.1 Enumerative Combinatorics

Changes:

• The function BernoulliNumber has been sped up very significantly by the use of power series (with asymptotically-fast arithmetic).

11.2 Incidence Structures and Designs

Changes:

• Most point and block functions now take an extra argument: the incidence structure in which the operation should be performed. This allows for more flexibility when working with incidence structures, and makes the incidence structure module more consistent with others such as groups.

New features:

• New function HadamardNormalize to normalize a Hadamard matrix to have only ones in the first row and first column.

• New function HadamardAutomorphismGroup to compute the automorphism group of a Hadamard matrix.

11.3 Finite Planes

Facilities for subplanes of finite projective and affine planes have been improved so as to make working with planes and subplanes much easier.

Changes:

• Most point and line functions now take an extra argument: the incidence structure in which the operation should be performed. This allows for more flexibility when working with planes and makes the module compatible with others.

• The following plane intrinsics have changed to accept the plane in which the operation should be performed as the first argument: AllPassants, AllSecants, AllTangents, CentralCollineationGroup, Conic, ContainsQuadrangle, Coordinates, Exterior, ExternalLines, Index, Interior, IsArc, IsCollinear, IsComplete, IsConcurrent, IsParallel, IsUnital, Knot, ParallelClass, Pencil, QuadraticForm, Support, Tangent, UnitalFeet.

11.4 Error-correcting Codes

New features:

• The function IsEquivalent has been documented for the first time.