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Magma V2.26 Release Notes

Release notes for Magma V2.26.

48 sections

Introduction

This document provides a terse summary of the new features released as part of Magma versions V2.26 (April 2021).

A small number of new features were exported in patch releases prior to the main release of V2.26 in April 2021 and these are also listed here for completeness. Only significant bugfixes are noted here – for a more complete list of bugfixes the reader should consult the patch release change log for V2.25-x.

Highlights

Arithmetic Geometry

Hyperelliptic Curves

The functionality for hyperelliptic curves and (smooth) plane quartic curves has been extended by R. Lercier, C. Ritzenthaler, and J. Sijsling. For these classes of curves \(C\), the following new features for hyperelliptic curves have been included:

  • The reconstruction of hyperelliptic curves of genus \(2\) (respectively \(3\)) from their Igusa invariants (respectively Shioda invariants) has been optimized. The use of separating invariants extends the reconstruction algorithms in genus \(3\) to all characteristics except \(5\). Moreover, for all automorphism groups, the resulting reconstructed curves are now returned over the base field whenever possible, and they have small coefficients when the base field is a number field.

  • Given two hyperelliptic curves \(C_1\) and \(C_2\) over a common base field, one can effectively check for the existence of isomorphisms \(C_1 \to C_2\) between them, both over the base field and over its algebraic closure. Moreover, the isomorphisms can be computed effectively.

  • Similarly, the automorphism group of a given curve \(C\) (over the base field and over the algebraic closure) can be computed. The automorphism group can be obtained as a permutation group.

  • Under some mild conditions on the characteristic, the twists of a given curve \(C\) over a finite field, that is, the isomorphism classes of curves that become isomorphic to \(C\) over the algebraic closure of the base field, can be calculated effectively.

Plane Quartic Curves

Some functionality for (smooth) plane quartic curves has been developed by R. Lercier, C. Ritzenthaler, and J. Sijsling similar to that which they have developed for hyperelliptic curves.

  • The Dixmier–Ohno invariants for smooth plane quartic curves (of genus \(3\)) have been implemented, as well as an optimized calculation of their discriminant.

  • An efficient algorithm for reconstructing a smooth plane quartic curve from its Dixmier–Ohno invariants is provided.

  • Given two plane quartic curves \(C_1\) and \(C_2\) over a finite field or the rationals, the existence of isomorphisms \(C_1 \to C_2\) can be determined, both over the base field and over its algebraic closure. Moreover, the isomorphisms can be effectively computed.

  • The automorphism group of a given curve \(C\) (either over the base field or over the algebraic closure) can be computed. The automorphism group can be obtained as a permutation group.

  • The twists of a plane quartic curve over a finite field can be computed.

Elliptic Curves Over Number Fields

  • Given an order \(O\) of a number field \(K\), the intrinsic EllipticCurveSearch attempts to find those elliptic curves that have \(O\) as conductor. A number of issues that emerged when \(K\) is a quartic field have been fixed.

Arithmetic Fields

\(p\)-adic Rings

  • A C version of Christopher Doris’s package for exact arithmetic in \(p\)-adic rings and fields is included in Magma for the first time.

Algebraic Number and Function Fields

  • In 1999, J. Montes introduced a new computational representation, the OM representation, for prime ideals in Dedekind domains. This approach for computing with fractional ideals in Dedekind domains has been applied to ideal computations in both number fields and algebraic function fields. A Magma package implementing the Montes approach for both number fields and function fields has been developed by Jens-Dietrich Bauch.

    Part of Bauch’s package was incorporated into Magma V2.25 and this has led to dramatic speed-ups in the computation of maximal orders and in the decomposition of primes in number fields and function fields.

    This release contains the \(OM\) representation functionality from Bauch’s package. This includes the machinery for ideals and divisors for both number fields and function fields. In particular, this makes possible the construction of ideals and divisors without the computation of maximal orders.

Basic Rings and Fields

Univariate Polynomial Rings

  • Polynomial factorization over finite fields has been greatly sped up when there is a very large number of factors of equal degree.

  • A number of additional speedups have been made which apply generally to the von zur Gathen/Kaltofen/Shoup-based algorithm for polynomial factorization over finite fields.

  • The &* operator, when applied to univariate polynomials of the same degree, has been greatly sped up.

  • A parallel version of the Schoenhage-Strassen FFT algorithm for multiplication of polynomials of high degree and with very large integer coefficients has been developed and may be selected with SetParallelFFT.

Multivariate Polynomial Rings

  • A major new sparse representation has been developed for monomials of multivariate polynomial rings. As a result, several types of computations involving polynomials with many variables have been sped up and take less memory (particularly when the monomials have very low degree). This includes basic arithmetic, the evaluation of homomorphisms, and the computation of Gröbner bases.

Coding Theory

Linear Codes over Finite Fields

  • POSIX threads and distributed computation are now available for the computation of the minimum weight of codewords. The speed-up factor is approximately equal to the number of cores used.

  • Similar parallelism is available for closely related computations such as the construction of the set of minimum weight codewords.

  • Both POSIX threads and distributed computation type parallelisation can now be used for the computation of the weight distribution and weight enumerator of a linear code. The speed-up factor is close to the number of cores used.

Commutative Algebra

Ideal Theory and Gröbner Bases

  • The function GroebnerBasis when applied to an ideal or set/sequence of polynomials now has a second return value \(D\) which gives the sequence of degrees encountered in each step of the \(F_4\) algorithm (whether computing the easy GB before a conversion, or a direct F4 computation), when that is non-trivial.

  • The function EasyIdeal similarly now has a second return value \(D\) giving the sequence of degrees encountered when computing the easy basis (the third return value is now the isomorphism map which used to be the second return value).

  • Major improvements have been made to the algorithm for computing the Hilbert series or Hilbert polynomial of an ideal of a multivariate polynomial ring \(R\), particularly when there are a large number of variables in \(R\).

  • The algorithm for computing the dimension of an ideal of a multivariate ring \(R\) and a corresponding maximally independent set has been greatly improved, particularly when there are a large number of variables in \(R\).

  • The sparse case of the \(F_4\) algorithm for inputs defined over \(\mathrm{GF}(2^k)\) for \(1<k<32\) has been improved.

  • The dense \(F_4\) algorithm in ‘HFE’ mode has been greatly improved when POSIX threads are used. The 80-variable Patarin HFE Challenge 1 can now be solved in 28 seconds on a 4-core 3.6GHz Intel Core i7-7700 processor.

Invariant Theory

  • Some very major improvements in speed and memory usage have been achieved in the linear algebra method for computing the monomials of given degree for an invariant ring (by use of sparse linear algebra where possible).

    This makes possible the computation of properties such as the primary and secondary invariants in the modular case for much larger invariant rings than was previously possible.

    For example, computing the primary invariants of the natural degree-5 representation of SU\(_5(2)\) (with coefficient ring \(\mathrm{GF}(2^2)\)) is now about \(40\) times faster than previously.

Group Theory

Classical Groups

  • The algorithm for the conjugacy invariants of classical groups has been improved and as a result there is a significant speed increase in computing the conjugacy classes of classical groups in odd characteristics.

  • Code from Eamonn O’Brien and Giovanni de Franceschi has been installed. Their package computes the conjugacy classes, centralisers and conjugating elements for all classical groups, including groups in characteristic 2.

  • The intrinsic Classes (and hence its synonym ConjugacyClasses) has been modified so that it uses the new new code when appropriate.

  • The implementation of the orthogonal groups of minus type in even dimensions (\(GO^-(2n,q)\) and subgroups) has been changed. For several years, even though the groups produced in different sessions of Magma were isomorphic, the generators were not necessarily the same. In order to address this problem the original C code for the generators has been restored.

Finite Groups

  • The new function MinimalDegreePermutationRepresentation developed by Derek Holt takes a finite group \(G\) (given as a permutation, matrix, abelian or pc-presentation group) as input, and computes a faithful permutation representation of \(G\) of smallest possible degree. The representation and its image, which is isomorphic to \(G\), are returned.

    This calculation can take a long time for some groups \(G\), even when \(G\) has faithful permutation representations of moderately small degree. There is an optional parameter Accept, which can be set by the user to a positive integer \(n\), in which case the function will stop as soon as it has found some faithful representation of degree at most \(n\) and output that. This can speed things up, but of course the result is not guaranteed to have the smallest possible degree in that case.

    The function works quickly on almost all of the groups in the small groups library of groups of order less than 2,000, but it can be slow on a few such examples, particularly large \(p\)-groups. These might be suitable candidates for effective use of the Accept parameter.

    It generally performs well with matrix groups, provided that a reasonably small degree faithful permutation representation exists. For example, for the group SpinPlus(8,3), it finds the optimal representation of degree 4,320 reasonably quickly (about 20 seconds), whereas for SpinMinus(8,3), it takes longer (about 180 seconds) to find the representation of degree 183,680 and prove that it has minimal degree.

    Although the new function cannot be used directly on a finitely presented group \(F\), the user can attempt to find an isomorphic permutation group using coset enumeration on \(F\) and then apply the minimal degree function.

  • A database containing the 403,874 simple groups of order less than \(10^{20}\) has been installed. The simple groups are arranged in increasing group order thereby making it straightforward for the user to loop though them. The groups can be accessed either by specifying their standard name or their position in the database. While Magma contains tools that allow the user to construct any finite simple group the purpose of this feature is to make it easier for users to access (the smaller) simple groups and to loop through a series of them.

Matrix Groups

  • A new update of the CompositionTree package is included in this release. The handling of representations of \(SL(2, q)\) inside CompositionTree was revised to ensure that all possible reductions are performed inside CompositionTree before calling RecogniseSL2; this improves performance for reducible representations of these groups. The intrinsic CompositionTreeVerify was revised so that all generators of the input group \(G\) are verified to lie in the group \(H\) described by the data structure constructed by CompositionTree. This additional check is needed to ensure that \(H = G\), rather than being just a subgroup of \(G\).

Soluble Groups

  • A package known as TameGenus was constructed by Peter Brooksbank, Josh Maglione, and James Wilson as an application of their Multilinear Algebra package. It includes polynomial-time algorithms for deciding isomorphism, constructing automorphism groups, and building canonical labels for a class of finite \(p\)-groups of nilpotency class \(2\) and exponent \(p\). This class includes all \(d\)-generator groups of order \(p^{d + 2}\), exponent \(p\), and class \(2\).

Integral Lattices

Enumeration in Lattices

  • Both POSIX threads and distributed computation type parallelisation are now available for the enumeration of shortest vectors. The speed-up factor achieved is approximately equal to the number of cores used.

  • The parallelisation of short vector enumeration automatically parallelises the computation of lattice minimum, packing radius, Hermite constant, centre density, density of the lattice and other computations that involve this type of enumeration.

  • Both POSIX threads and distributed computation type parallelisation are now available for the enumeration of closest vectors. The speed-up factor achieved is close to the number of cores used.

  • The parallelisation of closest vectors enumeration automatically parallelises computations that involve this type of enumeration.

Linear Algebra and Module Theory

Linear Algebra over Finite Fields

  • The algorithms to compute the semilinear dual, the twisted dual and the twisted semilinear dual of a \(G\)-module have been updated and as a result there is a considerable speed improvement.

  • The optimal Wilson algorithm for the Gram-Schmidt reduction of the matrix of a reflexive bilinear or sesquilinear form to (almost) diagonal form has been installed.

Multilinear Algebra

  • The calculation of invariant bilinear forms is no longer restricted to finite fields.

  • An upgrade of the package for Multilinear Algebra developed and maintained by Peter Brooksbank, Josh Maglione and James Wilson has been installed. In this upgrade some functions have been optimized and a number of bugs are fixed.

Parallelism

  • Several new parallelism facilities are included in Magma V2.26 which are explained in detailed in a new chapter of the Handbook entitled ‘Parallelism’ (Chapter 5). These include the following:

    • The support for user-implemented distributed parallelism has been greatly expanded.

    • POSIX threads and distributed type parallelisation are now available for the enumeration of short vectors and related properties.

    • POSIX threads and distributed type parallelisation are now available for the enumeration of close vectors.

    • The dense \(F_4\) algorithm in ‘HFE’ mode has been greatly improved when POSIX threads are used. The 80-variable Patarin HFE Challenge 1 can now be solved in 28 seconds on a 4-core 3.6GHz Intel Core i7-7700 processor.

    • Multiple NVIDIA GPUs are now supported for matrix multiplication over very small finite fields.

    • POSIX threads and distributed type parallelisation are provided for the computation of the minimum weight and weight distribution of a linear code.

Representation Theory

\(KG\)-Modules

  • There is a new function AbsolutelyIrreducibleModules(G) for a finite group \(G\) to compute all the non-isomorphic absolutely irreducible \(G\)-modules in characteristic zero, which are written over \(\mathbb{Q}\) or a number field of minimal degree. As an example, the 72 absolutely irreducible \(G\)-modules for the group PSL\((3, 8)\) are computed in about 2 hours, including dimension-511 modules written over a number field of degree 18.

  • The function IrreducibleModules(G, K) now supports the cases when \(K\) is a quadratic field or a simple number field and returns the non-isomorphic irreducible \(K[G]\)-modules of a finite group \(G\).

  • The function GModule(chi) to compute an individual absolutely-irreducible \(G\)-module for an absolutely irreducible character chi has been greatly improved for a wide class of inputs.

  • The functions that construct all irreducible and absolutely irreducible \(KG\)-modules, where \(K\) is a finite field and \(G\) is a non-soluble group, have now been extended to include matrix groups over all types of ring. Formerly, it was essentially restricted to permutation groups.

  • The functions that construct all irreducible and absolutely irreducible \(KG\)-modules modules, where \(K\) is a finite field and \(G\) is a soluble group, have been made much more reliable by using a new implementation of the Schur algorithm. (While the former implementation of the Schur algorithm is more general and faster for large groups, it has problems that cause it to fail sometimes).

System

NVIDIA GPU support

  • New functions to control usage of multiple NVIDIA GPUs when present.

  • Multiple NVIDIA GPUs are now supported for matrix multiplication over very small finite fields.

Documentation

New Handbook Chapters:

  • Parallelism

  • Plane Quartics: Invariants and Isomorphisms

Language and System Features

New Features:

  • New functions SetNGPUs, GetNGPUs, SetGPUDevices, SetGPUDevice to control usage of multiple NVIDIA GPUs when present.

  • When Magma is started from the shell, the command line option -t N may now be used to set the number of POSIX threads to \(N\) at startup, where \(N\) is a positive integer (so this is equivalent to calling SetNthreads(N); after Magma has started).

Aggregates and Mappings

Bug Fixes:

  • Map application has been fixed so that automatic coercion into the domain is avoided when the input is a sequence of elements in the domain itself. (V2.25-5)

Algebraic Geometry

Schemes

New Features:

  • It is now possible to construct a map between a projective scheme and an affine scheme with one of the defining polynomials being zero. (V2.25-5)

Bug Fixes:

  • A bug in the intrinsic Reduction which computes the reduction of a variety \(X\) over a number field \(K\) at a place or a prime ideal of \(K\) has been fixed. Reported with fix by Raymond van Bommel.

  • An error in PointSearch which caused a crash when applied to an affine scheme has been fixed.

  • A crash in IsInvertible applied to a map between schemes when the parameter Maximal is set to true has been fixed. (V2.25-6)

  • An error in the Faugere algorithm which caused the intrinsic PointsOverSplittingField (and others) when applied to a scheme to crash has been fixed.

Algebraic Surfaces

Bug Fixes:

Toric Varieties

Changes:

  • Maps involving toric varieties have been improved. (V2.25-6)

Arithmetic Geometry

Rational Curves and Conics

Bug Fixes:

  • An occasional crash in HasRationalPoint for conics defined over a number field has been fixed.

Elliptic Curves over the Rational Field

Bug Fixes:

  • An error in the intrinsic MordellWeilGroup caused by a recent change has been fixed.

Elliptic Curves over Number Fields

Bug Fixes:

  • A number of bugs that caused the intrinsic EllipticCurveSearch to crash sometimes when the conductor argument is an order in a quartic number field have been fixed.

Hyperelliptic Curves

In this subsection new intrinsics for hyperelliptic curves are summarised. These were made available in the 2020 version of a package for hyperelliptic curves being developed by R. Lercier, C. Ritzenthaler, and J. Sijsling.

Invariants

New Features:

Automorphisms and Isomorphisms

New Features:

Twists

New Features:

  • Twists: Given an elliptic curve, hyperelliptic curve or plane quartic curve, this intrinsic returns the twists of the curve.

  • TwistsOfHyperellipticPolynomials: Let \(C\) be the hyperelliptic curve defined by \(y^2 = f(x)\). Given \(f\) as argument, this intrinsic returns the twisted hyperelliptic polynomials and their geometric reduced automorphism groups.

  • TwistsOfHyperellipticPolynomials: Let \(C\) be the hyperelliptic curve defined by \(y^2 + h(x)y = f(x)\). Given \([f, g]\) as argument, this intrinsic returns the twisted hyperelliptic polynomials and their geometric reduced automorphism groups.

  • TwistedHyperellipticPolynomialsFromShiodaInvariants: Let \(C\) be the hyperelliptic curve defined by \(y^2 = f(x)\) and let \(SI\) be the Shioda invariants for \(C\). Given \(SI\) as argument this intrinsic returns the twisted hyperelliptic polynomials and their automorphism groups.

  • TwistedHyperellipticPolynomialsFromShiodaInvariants: Let \(C\) be the hyperelliptic curve defined by \(y^2 + h(x)*y = f(x)\) and let \(SI\) be the Shioda invariants for \(C\). Given \(SI\) as argument this intrinsic returns the twisted hyperelliptic polynomials and their automorphism groups.

  • TwistsFromShiodaInvariants: Given the Shioda invariants of a hyperelliptic curve \(C\) return the associated twisted hyperelliptic curves and their automorphism groups.

  • TwistsFromIgusaInvariants: Given the Igusa invariants of a hyperelliptic curve \(C\) return the associated twisted hyperelliptic curves and their automorphism groups.

  • TwistsFromG2Invariants: Given the Cardona-Quer-Nart-Pujolas invariants of a hyperelliptic curve \(C\) of genus \(2\) return the associated twisted hyperelliptic curves and their automorphism groups.

Reduced Isomorphisms and Automorphisms

New Features:

  • IsGL2EquivalentExtended: Given homogeneous polynomials \(f_1\) and \(f_2\) of degree \(n\) where \(n\) is at least \(4\) the intrinsic returns true if a matrix \(T\) exists such that the change of variable induced on \(f_1\) by \(T\), that is \(f_1*T\), is a multiple of \(f_2\). If there is such a matrix \(T\) then all such matrices are returned.

  • IsReducedIsomorphicHyperellipticCurves: Let \(C_1\) and \(C_2\) be two hyperelliptic curves with defining polynomials \(f_1\) and \(f_2\). Given \(C_1\) and \(C_2\) as arguments, the intrinsic returns a boolean indicating whether a matrix \(T\) exists that induces an isomorphism \(f_1(x) \rightarrow f_2(x)\). f there is such a matrix \(T\) then all such matrices are returned.

  • IsReducedIsomorphicHyperellipticCurves: Let \(C_1\) and \(C_2\) be two hyperelliptic curves with defining polynomials \(f_1\) and \(f_2\). Given \(f_1\) and \(f_2\) as arguments, the intrinsic returns a boolean indicating whether a matrix \(T\) exists that induces an isomorphism \(f_1(x) \rightarrow f_2(x)\). f there is such a matrix \(T\) then all such matrices are returned.

  • ReducedIsomorphismsOfHyperellipticCurves: Let \(C_1\) and \(C_2\) be two hyperelliptic curves with defining polynomials \(f_1\) and \(f_2\). Given \(C_1\) and \(C_2\) as arguments, the intrinsic returns a full list of matrices \(T\) that induce an isomorphism \(\phi f_1(x) \rightarrow f_2(x)\).

  • ReducedIsomorphismsOfHyperellipticCurves: Let \(C_1\) and \(C_2\) be two hyperelliptic curves with defining polynomials \(f_1\) and \(f_2\). Given \(f_1\) and \(f_2\) as arguments, the intrinsic returns a full list of matrices \(T\) that induce an isomorphism \(\phi f_1(x) \rightarrow f_2(x)\).

  • ReducedAutomorphismsOfHyperellipticCurve: Given a hyperelliptic curve \(C\), the automorphism group of the defining polynomial of \(C\) is returned in the form of a list of matrices.

  • ReducedAutomorphismGroupOfHyperellipticCurve: Given the hyperelliptic curve \(C\), defined by the equation \(y^2 = f(x)\), return the automorphism group of the polynomial \(f(x)\) (assumed to be of even degree), as a permutation group.

Point Counting

Bug Fixes:

  • An error that occurred sometimes in the intrinsic IsLocallySolvable when searching for points on a hyperelliptic curve has been fixed.

  • A memory leak in # for hyperelliptic curves has been fixed. (V2.25-8)

Plane Quartic Curves

The functionality for (smooth) plane quartic curves developed by R. Lercier, C. Ritzenthaler, and J. Sijsling is described here in detail.

Basic Operations

New Features:

Invariants

New Features:

Isomorphisms and Automorphisms

New Features:

Twists

New Features:

Arithmetic Fields (Global)

Algebraic Number Fields

New Features:

  • The verbose flag NumberFieldAlgorithms has been introduced to display when maximal orders, class groups and unit groups are computed as well as information about the orders they are being computed for.

Changes and Removals:

  • The implementations of the Round 2 algorithm, Buchmann–Lenstra and Montes algorithms for computing maximal orders have all been improved by using the Smith form of the trace matrix to attempt to compute primes dividing the discriminant more efficiently. (V2.25-6)

  • The implementation of the Buchmann-Lenstra algorithm has been further improved by the use of the Dedekind criterion. (V2.25-8)

  • Valuations of ideals of orders which are not known to be maximal is now computed by reducing the generators of the ideal by the prime ideal and its powers.

  • Compositum of two number fields has been made potentially more efficient by checking the smaller degree field for normality first. (V2.25-3)

  • Determining whether an element of a number field is an \(n\)th power has been made more efficient. (V2.25-5)

  • Checking for negative powers of zero has been added to PowerProduct.

Bug Fixes:

  • Several issues affecting the implementation of the Montes algorithm for computing maximal orders and decompositions of primes have been sorted out. (V2.25-8)

  • Using the map returned by IsIsomorphic applied to 2 fields when one field is non simple has been fixed. (V2.25-8)

  • Decomposition of ramified primes in non-maximal orders of Kummer and Artin–Schrier extensions has been fixed. (V2.25-7)

  • Error checking has been improved to meet for 2 number fields to ensure both are defined by a single polynomial. (V2.25-7)

  • A bug in coercion of elements in large towers has been fixed. (V2.25-6)

  • A bug in computing Automorphisms of a degree 2 field defined by a non-monic or non-integral polynomial has been fixed. (V2.25-7)

  • CoprimeRepresentative now always returns an element of a field of fractions of an order. (V2.25-4)

  • A bug in coercion of number field elements into coefficient rings in a tower has been fixed. (V2.25-5)

  • The LCM of zero and another element of a quotient of an order has been fixed to return zero. (V2.25-8)

  • A crash in the intrinsic Solutions when solving certain Thue equations has been fixed. (V2.25-9)

Quadratic Fields

Changes:

  • IsQuadratic now returns true for all degree 2 fields and also returns an isomorphic radical extension as a FldQuad. (V2.25-3)

Bug Fixes:

  • A bug in RingClassGroup for orders of quadratic fields has been fixed. (V2.25-7)

  • Conductors of orders of quadratic fields \(> 2^{30}\) are now handled better. (V2.25-2)

Class Field Theory for Number Fields

Changes:

  • Checking that extensions are normal and abelian when constructing an AbelianExtension has been increased. (V2.25-6)

  • The computation of RayClassGroups has been improved by the use of a power product representation of units.

Bug Fixes:

  • A crash in HeckeCharacterGroup due to equal but non identical orders has been fixed. (V2.25-9)

Algebraic Function Fields

Changes and Removals:

  • Valuations of ideals of orders which are not known to be maximal are now computed by reducing the generators of the ideal by the prime ideal and its powers.

  • The implementation of the Buchmann-Lenstra algorithm has been improved by the use of the Dedekind criterion. (V2.25-8)

Bug Fixes:

  • A bug when testing IsIsomorphic of two curves or function fields has been fixed by F. Hess. (V2.25-7)

  • Decomposition of ramified primes in non-maximal orders of Kummer and Artin–Schrier extensions has been fixed. (V2.25-7)

  • A bug in coercion of function field elements into coefficient rings in a tower has been fixed. (V2.25-5)

  • Several issues affecting the implementation of the Montes algorithm for computing maximal orders and decompositions of primes have been sorted out. (V2.25-8)

Galois Groups

Bug Fixes:

  • A fix has been made to the computation of the GaloisGroup of a reducible polynomial over a rational function field over \(\mathbf{Q}\) having zero as a root. (V2.25-7)

  • A fix has been made when the Prime parameter to GaloisGroup is set. (V2.25-7)

  • SolveByRadicals for polynomials over the rational field has has a recently introduced bug fixed. (V2.25-6)

  • The roots returned by GaloisSplittingField when given a polynomial over a number field were sometimes wrong. This has been fixed. (V2.25-9)

Montes Algorithm

The new ideal type OMIdl is available for ideals of number fields and function fields. The new divisor type OMDiv is available for function fields.

Changes and Removals:

  • The implementations of the Montes algorithm for computing maximal orders of number fields has been improved by using the Smith form of the trace matrix to attempt to compute primes dividing the discriminant more efficiently. (V2.25-6)

Bug Fixes:

  • Several issues affecting the implementation of the Montes algorithm for computing maximal orders and decompositions of primes have been sorted out. (V2.25-8)

New Features:

  • The Montes can be called directly for a polynomial or an algebraic field and a prime. The Single Factor Lifting Algorithm can also be called directly on a prime ideal in OM-representation using the intrinsic SFL.

  • The intrinsic SetUseMontes controls whether the Montes algorithm is used by default for maximal order and decomposition calculations in number fields and function fields. Information regarding such setting can be retrieved through GetUseMontes.

Ideals in OM-representation

New Features:

  • Ideals in OM-representation can be converted to an ideal of type RngOrdFracIdl by calling Ideal. An OM-representation can be computed from a RngOrdFracIdl or from generating elements. using OMRepresentation.

  • The sum, product, quotient and powers of ideals in OM-representation can be calculated.

  • Ideals in OM-representation can be tested for being the one or zero ideal and for equality with other ideals so represented. An element can be tested for being in an OMIdl. It can be determined whether an OMIdl is a subset of another such ideal.

  • Primality and integrality of ideals in OM-representation can be determined.

  • A pIntegralBasis, SIntegralBasis and Basis can be computed for an ideal in OM-representation. Two elements which generate an ideal in OM-representation are turned by TwoElement.

  • Elements and OM-represented ideals can have their Valuation calculated at a prime ideal in OM-representation. Elements can be reduced mod such prime ideals and a local expansion computed using Reduction.

  • The Factorization of ideals in OM-representation can be calculated.

  • Prime ideals in OM-representation can have their ResidueField and Degree computed.

Divisors in OM-representation

New Features:

  • Divisors in OM-representation can be constructed from one or two OMIdl or a function field element using OMDivisor.

  • The ZeroDivisor and PoleDivisor of function field elements are divisors in OM-representation.

  • A divisor of degree one of a function field can be constructed using OMDivisorOfDegreeOne.

  • OM-representations can be computed for divisors of type DivFunElt using OMDivisor and Divisors of type DivFunElt can be formed from OMDiv.

  • Divisors in OM-representation can be added, subtracted and multiplied by a scalar. The GCD of 2 divisors can be computed.

  • Equality of divisors in OM-representation is available. Divisors can be tested for being positive (IsPositive) and principal (IsPrincipal).

  • The Support of a divisor in OM-representation is a sequence of prime ideals in OM-representation and a sequence containing the valuations of the divisor at those primes. The Valuation of a divisor can be computed at prime ideals in OM-representation.

  • Divisors in OM-representation can have their Height and Degree computed.

  • A Basis for the Riemann-Roch space of a divisor in OM-representation can be computed along with the Dimension of the space. A ReducedBasis and a SemiReducedBasis are also available.

  • The SuccessiveMinima and ApproximatedSuccessiveMinima of a divisor can be calculated.

Arithmetic Fields (Local)

\(p\)-adic Rings and their Extensions

Changes and Removals:

  • The computation of GCDs during Factorization of polynomials over local rings has been made more efficient when one of the polynomials is linear and divides the other.

Bug Fixes:

  • A bug affecting pSelmerGroup, PrincipalUnitGroupGenerators and ResidueSystem for local rings containing more than 1 unramified extension has been fixed. (V2.25-6)

  • A bug in the construction of a local field which is not a scalar multiple of a monic integral polynomial has been fixed. (V2.25-6)

  • Testing IsNormal for finite precision rings has been fixed. (V2.25-4)

Series Rings

Changes and Removals:

  • The computation of GCDs during Factorization of polynomials over local rings has been made more efficient when one of the polynomials is linear and divides the other.

  • Improvements have been made to GCD computations for polynomials over extensions of series rings.

Bug Fixes:

Basic Rings and Fields

Integer Ring

Bug Fixes:

  • A rare crash in ECPP (coming from IsPrime) has been fixed (V2.25-9).

Finite Fields

New Features:

  • The default strategy for selecting the Pohlig-Hellman discrete logarithm algorithm has been improved for general finite fields. Also, if \(K\) is a finite field, one may now set the attribute

        K`UsePHLog

    (with the backquote operator) to assert whether the Pohlig-Hellman algorithm should or not should be used when computing logarithms in \(K\).

Changes and Removals:

  • Basis has been fixed to be a relative basis. (V2.25-6)

Real and Complex Fields

Bug Fixes:

  • A bug in computing the complex roots of a polynomial has been fixed. (V2.25-2)

Univariate Polynomial Rings

New Features:

  • Polynomial factorization over finite fields has been greatly sped up when there are a very large number of factors of equal degree.

  • The von zur Gathen/Kaltofen/Shoup-based algorithm for polynomial factorization over finite fields has had some further speedups in general.

  • The &* operator, when applied to univariate polynomials of the same degree, has been greatly sped up.

  • A parallel version of the Schoenhage-Strassen FFT algorithm for multiplication of polynomials of high degree and with very large integer coefficients has been developed and may be selected with SetParallelFFT.

Bug Fixes:

  • A bug in Normalize for univariate polynomials defined over integer residue class rings has been fixed, so the result is guaranteed to be unique. (V2.25-6)

  • A bug in EquidimensionalPart has been fixed (V2.25-5).

  • A bug that caused a crash in the intrinsic Factorization when applied to a univariate polynomial over a large degree (32) number field has been fixed.

Multivariate Polynomial Rings

New Features:

  • A major new sparse representation has been developed for monomials of multivariate polynomial rings. As a result, several types of computations involving polynomials with many variables have been sped up and take less memory (particularly when the monomials have very low degree). This includes basic arithmetic, the evaluation of homomorphisms, and the computation of Gröbner bases.

  • The new procedure SetSparseMonomialMinRank(R) sets the limit \(R\) so that from now on, whenever a multivariate polynomial ring \(P\) is created with at least \(R\) variables (i.e., has rank at least \(R\)), the sparse representation is used for the monomials of \(P\). The initial value of \(R\) is currently 20. Setting \(R\) to 0 means that the sparse representation will not be used (so the existing dense representation alone will be used). The corresponding function GetSparseMonomialMinRank() returns the current value of \(R\).

  • Some automatic coercions are now handled properly with chains of mixed univariate/polynomial rings.

Bug Fixes:

  • A crash when computing the variety of a zero-dimensional bivariate ideal over the algebraic closure of a finite field has been fixed.

  • The check for overflow in the number of monomials in a multivariate polynomial ring was not quite correct and has been fixed. (V2.25-6)

  • Crashes in multivariate polynomial GCD over finite fields have been fixed. (V2.25-7, V2.25-9)

  • A hang in polynomial GCD over a certain class of high-degree number fields has been fixed. (V2.25-7)

Coding Theory

Linear Codes over Finite Fields

New Features:

  • Both POSIX threads and distributed computation are now available for the computation of the minimum weight and weight distribution of a code. The speed-up factor achieved is approximately equal to the number of cores used. Many associated operations such as the enumeration of all codewords of minimum weight also benefit.

Bug Fixes:

  • A crash in ConstantWords over large non-prime finite fields has been fixed. (V2.25-8)

Commutative Algebra

Ideal Theory and Gröbner Bases

New Features:

  • The function GroebnerBasis when applied to an ideal or set/sequence of polynomials now has a second return value \(D\) which gives the integer sequence of degrees encountered in each step of the \(F_4\) algorithm (whether computing the easy GB before a conversion, or a direct F4 computation), when that is non-trivial.

  • The function EasyIdeal similarly now has a second return value \(D\) giving the sequence of degrees encountered when computing the easy basis (the third return value is now the isomorphism map which used to be the second return value).

  • The algorithm to compute the Hilbert series or Hilbert polynomial of an ideal of a multivariate ring \(R\) has been greatly improved, particularly when there are a large number of variables in \(R\).

  • The algorithm to compute the dimension of an ideal of an ideal of a multivariate ring \(R\) and a corresponding maximally independent set has been greatly improved, particularly when there are a large number of variables in \(R\).

  • The \(F_4\) algorithm for inputs defined over \(\mathrm{GF}(2^k)\) for \(1<k<32\) in the sparse case has been improved.

  • For the dense \(F_4\) algorithm in ‘HFE’ mode, the whole reduction phase is now completely parallelised when POSIX threads are selected by SetNthreads. This is faster than the distinct calls to matrix multiplication in the general dense algorithm and significant memory saving is also achieved. The 80-variable Patarin HFE Challenge 1 can now be solved in 28 seconds on a 4-core 3.6GHz Intel Core i7-7700 processor.

Changes:

  • The second return value of GroebnerBasis(S) for a set/sequence \(S\) is now the third return value and the second return value of EasyIdeal is now the third return value; see above for more details.

Bug Fixes:

  • A crash when computing large Groebner bases over non-prime finite fields of characteristic 2 has been fixed. (V2.25-6)

  • Incorrect sign handling when computing coordinates of polynomial/vectors in ideals/modules defined over exterior algebras has been fixed. (V2.25-6)

  • A missing error check for noncommutative Groebner basis computation over unsupported rings has been added. (V2.25-6)

  • A bug in Variety for ideals defined over algebraic closures has been fixed. (V2.25-8)

  • A crash in computing Groebner bases over the field GF(\(2^k\)) for \(k=2,3,4\) has been fixed. (V2.25-9)

  • A crash in computing primary decomposition over algebraically closed fields has been fixed. (V2.25-9)

Modules over Multivariate Polynomial Rings

New Features:

  • The algorithm to compute the Hilbert series or Hilbert polynomial of a module over a multivariate ring \(R\) has been greatly improved, particularly when there are a large number of variables in \(R\).

Invariant Theory

New Features:

  • Some very major improvements in speed and memory usage have been achieved in the linear algebra method for computing the monomials of given degree for an invariant ring (by use of sparse linear algebra where possible).

    This now allows the computation of properties such as the primary and secondary invariants in the modular case for much larger invariant rings than were previously possible.

    For example, computing the primary invariants of the natural degree-5 representation of SU\(_5(2)\) (with coefficient ring \(\mathrm{GF}(2^2)\)) is now about 40 times faster than previously.

Groups

Classical Groups

New Features:

  • The algorithm to compute the conjugacy invariants of all classical groups in odd characteristic has been updated and as a result there is a significant speed increase in computing the conjugacy classes of classical groups.

    Thanks to new code from Eamonn O’Brien and Giovanni de Franceschi it is now possible to compute the conjugacy classes, centralisers and conjugating elements for all classical groups, including groups in characteristic 2. The new intrinsics include ClassicalIsConjugate, ClassicalCentraliser, ClassicalClassMap, ClassesForFixedSemisimple and IsometryGroupClassLabel.

Changes:

Bug Fixes:

  • Two bugs in the code for GLNormaliser which caused it to crash have been fixed.

Finite Groups

Additions:

  • The new intrinsic MinimalDegreePermutationRepresentation developed by Derek Holt takes a finite group \(G\) (of type GrpPerm, GrpMat, GrpAb or GrpPC) as input, and computes a faithful permutation representation of \(G\) of smallest possible degree. The representation and its image, which is isomorphic to \(G\), are returned.

Matrix Groups Over Finite Fields

Changes:

  • The handling of representations of \(SL(2, q)\) inside CompositionTree has been revised to ensure that all possible reductions are performed before calling RecogniseSL2.

  • The intrinsic CompositionTreeVerify was revised to ensure that all generators of the input group \(G\) lie in the group \(H\) defined by the CompositionTree data structure.

  • The code which converted the return value of tt SpinorNorm to the Zassenhaus convention has been removed. This ensures that in characteristic not 2 the spinor norm of an element \(g\) of the orthogonal group is consistent with SpinorNorm(V,g), where \(V\) is a quadratic or inner product space.

\(p\)-Groups

New Features:
The TameGenus package developed by P.A. Brooksbank, J. Maglione, and J.B. Wilson provides fast algorithms for computing automorphism groups and deciding isomorphism for “tame genus” \(p\)-groups.

Databases of Groups

New Features:

A database which currently comprises the 403,874 simple groups of order less than \(10^{20}\) has been installed. While Magmacontains tools that allow the user to construct any finite simple group, the purpose of this facility is to make it easy to iterate through a selection of simple groups. When reading the descriptions of the intrinsics below it is important to note that a simple group may be specified in one of three ways:-

  • A string giving the standard name of the simple group.

  • An internal description of the simple group consisting of a tuple containing three integers.

  • The number of the simple group in the database.

  • The intrinsic SimpleGroup returns a simple group isomorphic to the specified simple group. The group may be specified by any one of the above three methods.

  • The intrinsic SimpleGroupOfOrder returns a simple group of the indicated order. The order may be given either as an integer or as a factorisation sequence containing the factored order.

  • The intrinsic NumberOfSimpleGroups returns the number of simple groups present in the database.

  • Given the database number of a simple group the intrinsic SimpleGroupName returns the standard name of the simple group.

  • Given the standard name of a simple group the intrinsic SimpleGroupNameToNumber returns the number of the simple group in the database.

  • Given the internal descriptor of a simple group the intrinsic SimpleGroupIdToNumber returns the number in the database of the simple group specified by the internal descriptor.

  • Given the database number of a simple group the intrinsic SimpleGroupId returns the internal descriptor as a tuple of three integers.

Lattices and Quadratic Forms

Lattices

New Features:

  • Both POSIX threads and distributed computation type parallelisation are now available for the enumeration of shortest vectors. The speed-up factor achieved is approximately equal to the number of cores used.

  • The parallelisation of short vector enumeration automatically parallelises the computation of lattice minimum, packing radius, Hermite constant, centre density, density of the lattice and other computations that involve this type of enumeration.

  • Both POSIX threads and distributed computation type parallelisation are now available for the enumeration of closest vectors. The speed-up factor achieved is close to the number of cores used.

  • The parallelisation of closest vectors enumeration automatically parallelises computations that involve this type of enumeration.

  • The integral variant of the LLL algorithm (selected by the parameter Method := "Integral") now has POSIX-thread support (selected by SetNthreads).

  • The algorithm for Coordinates for lattice elements defined over a real field has been made more robust.

  • New functions ShortVector, ShortestVector, CloseVector, ClosestVector to return a single non-zero short/close/shortest/closest vector in a lattice.

Bug Fixes:

  • A bug giving incorrect results in the maps for DualQuotient has been fixed. (V2.25-3)

  • Setting one of the attributes Minimum, Kissing Number, or Theta Series when one of the other attributes is set will no longer print the theta series. (V2.25-6)

  • Bugs in IsCoercible and Coordinates for lattice elements defined over a real field have been fixed.

Lie Algebras

New Features:

  • The closure algorithm for the computation of subalgebras of Lie algebras has been greatly improved, particularly in the case of large sparse algebras in characteristic zero.

Linear Algebra and Module Theory

Matrices

New Features:

  • Multiple NVIDIA GPUs are now supported for matrix multiplication over very small finite fields.

Changes:

Bug Fixes:

  • An issue where the multiplication of two large integer matrices could incorrectly compute the resulting size (leading to memory exhaustion) has been fixed. (V2.25-4)

  • A bug when computing the determinant of matrices over polynomial rings over non-domains has been fixed. (V2.25-6)

  • A crash in the integer Hermite algorithm (arising from computing the map arising from calls by AbelianQuotient) has been fixed. (V2.25-7)

  • A bug that occurred when taking projections of subspaces of a vector space over the rationals has been fixed. (V2.25-7)

New Features:

  • The efficiency of the function Submatrix(S, I, J) where \(S\) is a sparse matrix and \(I,J\) are index sequences has been greatly improved (by writing an inefficient internal version).

Bilinear and Sesquilinear Forms

New Features:

  • The intrinsic GramSchmidtPair performs a Gram-Schmidt reduction of the matrix of a reflexive bilinear form using a new optimal algorithm due to James Wilson.

  • The intrinsic IsReflexive determines whether a matrix represents a reflexive form; if it does, the type of the form and the multiplier are returned.

  • EnsureUpperTriangular returns returns the upper triangular matrix defining the same quadratic form as a given matrix.

Changes:

  • The number of values accepted by the Variant parameter of the intrinsics StandardQuadraticForm and StandardSymmetricForm has been reduced from three to two: "Default" and "Revised". This is for consistency with the changes to the generators of the even dimensional orthogonal groups of minus type. The "Revised" variant returns the form preserved by the groups constructed by AltGOMinus, AltSOMinus and AltOmegaMinus.

Vector Spaces

New Features:

  • The := operator has been improved for vector space elements.

Multilinear Algebra

New Features:

Linear Associative Algebras

Associative Algebras

Bug Fixes:

  • Incorrect printing of elements in Magma mode has been fixed.

Quaternion Algebras

Bug Fixes:

  • A bug when computing of the discriminant of a quaternion algebra has been fixed by J. Voight. (V2.25-6)

Representation Theory

\(K[G]\)-Modules

New Features:

  • There is a new function AbsolutelyIrreducibleModules(G) for a finite group \(G\) to compute all the non-isomorphism absolutely irreducible \(G\)-modules in characteristic zero, which are written over \(\mathbb{Q}\) or a number field of minimal degree. As an example, the 72 absolutely irreducible \(G\)-modules for the group PSL\((3, 8)\) are computed in about 2 hours, including dimension-511 modules written over a number field of degree 18.

  • The function IrreducibleModules(G, K) now supports the case that \(K\) is a quadratic field or a simple number field and returns the non-isomorphic irreducible \(K[G]\)-modules of a finite group \(G\).

  • The function GModule(chi) to compute an individual absolutely-irreducible \(G\)-module for an absolutely irreducible character chi has been greatly improved for a wide class of inputs.

  • The functions IrreducibleModules(G, K) and AbsolutelyIrreducibleModules(G, K), where \(G\) is a non-soluble group and \(K\) is a finite field, have now been extended to include matrix groups over all types of ring. Formerly, it was essentially restricted to permutation groups.

Changes:\

There is an important change when computing irreducible \(KG\)-modules, where \(K\) is a finite field and \(G\) is a matrix group:

Bug Fixes:

  • A bug has been fixed which caused functions to compute irreducible modules for a group over a finite field (such as IrreducibleModules and AbsolutelyIrreducibleModules) to crash in certain circumstances. (V2.25-7)

  • A bug in the code for the function IrreducibleModulesSchur which occurs when computing \(KG\)-modules, \(K\) a finite field and \(G\) a soluble group, has been fixed. (V2.25-7)

  • A bug in the characteristic-0 Meataxe which caused it to hang in the case of some small \(KG\)-modules has been fixed. (V2.25-7)

  • A bug in the characteristic-0 Meataxe which caused it to crash when splitting a module over a cyclotomic field has been fixed. (V2.25-7)

  • An incorrect result for the function IndecomposableSummands (or DirectSumDecomposition) applied to a \(\mathbb{Q}G\)-module where G is a soluble group has been fixed. (V2.25-8)

  • A crash in IsIsomorphic for \(G\)-modules over finite fields has been fixed. (V2.25-9)

  • The IsIsomorphic algorithm for \(G\)-modules over finite fields has been improved, so that the initial tests for non-isomorphism are much stronger, fixing slow behaviour for certain inputs. (V2.25-9)

  • A crash in intrinsic IrreducibleModules when applied to a \(KG\)-module, where \(K\) is the rationals, has been fixed. (V2.25-9)

  • A crash in intrinsic Constituents when applied to a \(KG\)-module, where \(K\) is a cyclotomic field, has been fixed. (V2.25-9)

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