This document provides a terse summary of the new features released as part of Magma versions V2.29 (September 2025).
A small number of new features were exported in patch releases prior to the main release of V2.29 and these are also listed here for completeness. For a complete list of bugfixes throughout V2.28, the reader should consult the patch release change logs for V2.28-n for n from 2 to 27.
New Handbook Chapters are the following:
Étale Algebras (42)
Reductive Groups (75)
Group Representations (102)
Algebraic Modular Forms (150)
A crash in Embed for finite fields (involving default and non-default fields) has been fixed. Reported by Sachi Hashimoto.
A bug in NewformDecomposition involving a class of newforms over number fields has been fixed. Reported by Begum Cakti.
Incorrect documentation for the function Discriminant for univariate polynomials has been fixed. Reported by Michael Zieve.
An error message from IsogenyFromKernel has been corrected which affected subgroups of elliptic curves over a p-adic field. Reported by Sachi Hashimoto.
A new 'Large' version of Magma has been developed, which is available as an optional executable download (currently for the 64-bit Linux avx2 version only).
The existing default 'Standard' 64-bit version of Magma (which will remain as the default version) has always used 64-bit operations for operations involving packed data (including large integers, matrices over small finite fields, permutations, etc.) and the total memory allowable or the size of any individual memory block has been fully 64-bit (with no 32-bit limit). But the Standard version uses 32-bit quantities for indexing generic Magma objects, so this means that it has certain limits on the number of generic Magma objects or the size of some types of Magma aggregate structures (such as sets and sequences).
In contrast, the new Large version of Magma uses 64-bit quantities in such contexts. Thus the following have 64-bit limits in the Large version (instead of 32-bit limits in the Standard version):
The number of memory blocks (in the Standard version, if that number is reached, Magma must abort immediately). In particular, this includes the number of large integers (integers which must be stored in memory blocks) which can be created.
The cardinality of sets, sequences and other aggregate structures.
The length of objects represented as vectors (including standard vectors over rings).
The number of rows/columns in matrices (both dense and sparse).
The size of the matrices in the linear algebra phase of the F4 Gröbner basis algorithm.
At the same time, 64-bit long integers are exploited more in the Large version in algorithms which use relatively small integers. In particular:
Integers with absolute value between 230 and 262 are not stored in a separate block (but become 'immediate' small [machine 64-bit long] integers), thus saving time and space for computations with integers in this range. A similar improvement is achieved for elements of finite fields with cardinality in this range, and similar types using integers in this range.
Algorithms for matrices or polynomials with entries from the ring of integers or a finite field which are up to 64-bits are sped up similarly.
PLEASE NOTE: the Large version may need significantly more total memory than the Standard version for many kinds of computations (because the indexing of unpacked generic Magma objects is changed from using 32-bit to 64-bit quantities), so that is why the Standard version remains the default for now. But the Large version will allow Magma to perform several types of computations which have been impossible up till now, and it overcomes several errors/failures which have been arising when 32-bit quantities overflow in the Standard version.
New Features:
When creating an associative array A, one may now specify a default value D (any constant value) to be associated with A, via the parameter Default, as follows:
A := AssociativeArray(: Default := D); // Null index universe
A := AssociativeArray(I: Default := D); // Index universe I
If a default value D has been specified, then whenever A[x] is used but is not yet defined (i.e., x is not in the keys of A), then D is returned, instead of an error being raised. This allows constructions like the following, where the values of A will be sequences which are to be appended to dynamically:
A := AssociativeArray(: Default := []);
x := 3; y := 5;
Append( A[x], y);
Here we can append an element to A[x] even when x is not yet in the keys of A; in such a case, A[x] is initially taken to be [] and so can be appended to without error.
New Features:
DirectProduct has been rewritten independently of the Toric Varieties for other ambients.
ChangeRing has been provided for maps between schemes.
Changes and Removals:
The grading of a ProjectiveSpace is now transferred to the coordinate ring even when that grading contains a zero entry. (V2.28-3)
A crash in PointSearch for ambient spaces has been fixed. (V2.28-8)
New Features:
The ZetaFunction intrinsic has been updated by the addition of an implementation of a p-adic algorithm developed by Kyng. This algorithm can be applied to any geometrically irreducible curve. This makes feasible the computation of zeta functions for many inputs for which there is no known method for computing a good lift for Tuitman's algorithm, and the new algorithm is now run by default for input plane models with genus exceeding 5 that are not nondegenerate.
Bug Fixes:
A hard coded limit in the preimage of the map from the function field into the algorithmic function field has been lifted. This limit interfered unnecessarily in the computation of a generating element for a principal divisor as tested by IsPrincipal.
New Features:
The intrinsic RationalPointsGenus2 has been implemented by Michael Stoll, which attempts to compute the set of rational points on a hyperelliptic curve of genus 2. It uses a combination of techniques: search for (relatively small) points, test for everywhere local solubility, two-cover descent, degree 2 and 3 elliptic subcovers of rank zero, computation of generators of a finite-index subgroup of the Mordell-Weil group, Chabauty when the rank is at most 1, a check whether the curve has rational divisors of odd degree, if the rank is at least 2 and no points were found, a Mordell-Weil sieve computation.
The printing of hyperelliptic curves has been improved in the case that the coefficient of the y-term is negative.
Changes and Removals:
The intrinsic ReduceModel has been rewritten by Michael Stoll, and its optional parameters have changed. In particular, the parameters Simple and Al have been removed. The parameter Smallest and Height have been added to control the type of reduction.
New Features:
The package for computing with Modular Symbols has been significantly sped up, particularly so that decomposition of spaces is often much faster than previously.
Features:
This package computes spaces of algebraic modular forms for definite orthogonal and unitary groups over an arbitrary totally real number field F, of weight which could be any finite dimesional representation of the group, and level O(Λ) (or U(Λ)) where Λ is any integral positive definite ℤF-lattice These spaces are created using AlgebraicModularForms, OrthogonalModularForms and UnitaryModularForms.
Computations are done, via transversing the p-neighbours in the genus of a lattice, following an algorithm due to Greenberg and Voight; this is done internally and does not feature in the interface.
The dimension of a space may be computed (which is a nontrivial computation, as dimension formulae are not available in general).
Hecke operators can be computed, with respect to some fixed basis of the space, consisting of characteristic functions of the chosen representatives for the genus. (Currently there are no conventions regarding the choice of representatives.)
The Decomposition of a space is available, as well as HeckeEigenforms, and systems of eigenvalues can be obtained for the modular form corresponding to each Hecke-irreducible component in the decomposition.
New Features:
Embeddings between number fields and orders of number fields can be retrieved using HasEmbedding. Whether an embedding can be computed between two fields or orders can be determined by CanComputeEmbedding.
Decomposition of finite places of coefficient fields in extensions has been enabled. (V2.28-9)
Changes and Removals:
The class group and unit group computation has undergone a thorough review. As a result, a new algorithm for the GRH-conditional residue of the Dedekind zeta function has been implemented. It can be called directly by ResidueGRH the error can be controled by ResidueGRHbound. Further, bugs in the saturation checks for class group and unit group were found and fixed. Using all implication of GRH, the proof levels of the class group have been reordered. Now, a GRH-conditional result is the lowest proof level. Without the specification of a proof level, the results are rigorous.
IsSubfield and IsIsomorphic now respect existing embeddings. In addition IsIsomorphic no longer installs an embedding so there is no coercion resulting between the fields and the map returned needs to be used to move elements between the fields.
Improvements to coercion between number fields and between orders of number fields have been made.
IsIsomorphic now chooses coefficient embeddings that lead to subfield and isomorphism relationships over those that do not. (V2.28-3)
Error checking has been improved for the ext constructor when the Abs parameter is true. (V2.28-4)
Bug Fixes:
Expressing a field as a RelativeField of itself has been fixed. (V2.28-4)
Extends now handles relative extensions properly. (V2.28-6)
A bug affecting the integrality of an IntegralBasis computed using the Montes algorithm has been fixed. (V2.28-6)
An "Ambiguous signature match" has been removed for IntegralBasis. (V2.28-6)
Fixes have been made to recent improvements in MaximalOrder computations. (V2.28-9)
New Features:
Embeddings between orders of function fields can be retrieved using HasEmbedding. Whether an embedding can be computed between two orders can be determined by CanComputeEmbedding.
Changes and Removals:
Decomposition of primes in extensions of a polynomial ring or valuation ring now uses the order given as the first argument to decompose into rather than decomposing into a maximal order.
Improvements have been made to MaximalOrder computations.
RationalFunction for elements of non simple function fields now ensures the element has known coefficients rather than returning 0. (V2.28-9)
Applying Automorphisms to elements with a product representation no longer computes coefficients of the element. (V2.28-9)
Taking preimages of the identity automorphism has also been fixed. (V2.28-9)
Bug Fixes:
A crash in working with elements of a quotient of an order of a function field has been fixed. (V2.28-8)
New Features:
Special support for polynomials of the shape f(x) = g(xd) has been added. To use is, the option Deflation has to be set in GaloisGroup.
If a Galois group is computed twice with respect to different primes, an isomorphism of the results can be computed by GaloisGroupConjugation.
Changes and Removals:
Precision handling has been improved in SplittingField so that the valuation of the discriminant can be accurately determined. (V2.28-9)
User provided primes using the Prime parameter to GaloisGroup are now checked to be unramified. If a user given prime is ramified, another prime will be selected for the computation. (V2.28-9)
Checking for intransitive groups has been added to GaloisQuotient. (V2.28-9)
Changes and Removals:
A polynomial must now retain its degree when mapped to the residue class field in order to be inertial.
Increased error checking of the map given as the second argument to RelativeField is now done. (V2.28-9)
Bug Fixes:
The map returned from AbsoluteTotallyRamifiedExtension has been fixed. (V2.28-4)
New Features:
Primality testing and proving has been greatly improved as follows:
For testing whether n is prime and n < B, where B = 3317044064679887385961981 (approx 3.3×1024), the deterministic Sorenson/Webster method (with proof) is now used (using Miller-Rabin with the first 12 primes as bases).
For testing whether n is prime and n < 264, a fast in-place Montgomery method is also now used, which is much faster than previously.
A new version of the elliptic curve prime proof has been implemented. It can handle 1000 digit numbers in a few minutes, which is at least 10 times faster than previously. The primality of numbers with more than 5000 digits has already been proved with it. The function PrimalityCertificate returns the primality certificate generated by the elliptic curve prime proof. It can be verified and displayed with CheckCertificate. For compatibility, OldCertificate can be used to convert the new certificate to the old format.
New Features:
Improvements have been made in the algorithms for the factorization of univariate polynomials over finite fields.
New Features:
Some major improvements have been made to the algorithm for computing multivariate resultants. This benefits in particular the Trager algorithm for factoring multivariate polynomials over number fields.
New Features:
Major new improvements have been achieved for computing Gröbner bases of multivariate ideals where the coefficients lie in an algebraic number field. The new methods work particularly well for number fields of high degree (for which there had previously been no efficient methods in Magma) but are also faster than previously for fields of small degree.
The critical algorithm for computing the primary decomposition of multivariate ideals has been improved in various ways, both for the zero-dimensional and positive-dimensional cases (involving quite different techniques and subalgorithms for each case). This algorithm is at the heart of decomposition of schemes and now allows the computation of some decompositions to be feasible which were not previously.
Features:
This package provides interface for reductive groups such as orthogonal, symplectic and unitary groups over a number field. Currently supports groups that are split extensions of their identity component (but not necessarily connected), and such that each of the simple factors in their almost simple factorization is either orthogonal or unitary.
Given a symplectic, orthogonal or unitary vector space over a numebr field, one can form the corresponding SymplecticGroup, OrthogonalGroup, or UnitaryGroup. One can also use ReductiveGroup to create a reductive group from a group of Lie type specifying its component of the identity, and a finite group specifying its component group.
One can also create elements in these groups, and perform simple arithmetic with such elements.
New Features:
A comma-separated list with N expressions is now allowed for the right-hand-side (RHS) of an assignment statement, when the number of lvalues on the left to be assigned equals N. The expressions on the RHS are all evaluated first before any assignment on the left. For example, the following statement can now be used to swap variables a and b:
a, b := b, a;
Note that for each expression on the RHS which has multiple return values, all those values are discarded except for the first. For example, the following statement will set a to 1 and b to 7 (only the first return value is used for each call of Quotrem):
a, b := Quotrem(13, 10), Quotrem(75, 10);
A major improvement has been done in the language interpreter so that reference arguments with indexing in procedure calls are handled much more efficiently, and undesired temporary cloning of objects is now avoided (e.g., in procedure calls involving the tilde operator before indexed references such as S[i], and operators like +:= with indexed references on the LHS).
A new function Clone(X) has been added, which takes an object X having user-defined type T and returns a copy C of X with type T and with all assigned attributes copied from X. This allows one to modify the attributes of the new object C without affecting the attributes of X.
New Features:
Lattices over number nields now support Hermitian lattices, by specifying the optional parameter Involution in the constructor NumberFieldLattice.
IsIsometric and AutomorphismGroup now support the optional parameter Proper, in order to considered only proper (orientation-preserving) isometries.
Genus now supports displaying ang generating genus symbols, with Genera for generating all genus symbols of given signature and determinant.
GenusRepresentatives of lattices nowuse the mass formula for early termination.
Changes and Removals:
WittInvariant has been modified to be consistent with number field lattices and with the literature. It is no longer equivalent to the HasseInvariant.
Bug Fixes:
A crash in IsGLZConjugate involving short vector enumeration has been fixed.
Fixed a bug in IsIsometric for non-free lattices.
New Features:
Major new improvements have been achieved for computing with matrices where the coefficients lie in an algebraic number field (this includes algorithms for computing nullspaces, echelon forms and determinants). The new methods work particularly well for number fields of high degree (for which there had previously been no efficient methods in Magma) but are also faster than previously for fields of small degree.
Changes:
A matrix M is diagonalisable if its Jordan form is diagonal; i.e., there is an invertible matrix T such that TMT-1 is diagonal. Alternatively a symmetric matrix is diagonalisable if it is the Gram matrix of an orthogonal basis; i.e., there is a matrix T such that TMT\hbox{tr} is diagonal.
To distinguish between these two types of diagonalisation, the intrinsic Diagonalization that applies to symmetric matrices has been renamed Orthogonalization.
The intrinsic which diagonalises a matrix in the sense of finding a diagonal Jordan form retains the name Diagonalization. Furthermore, the names Diagonalisation and Diagonalization are now synonyms.
There is a new intrinsic IsDiagonalisable (and IsDiagonalizable is a synonym). The default behaviour checks whether a matrix is diagonalisable over its field of definition. If the parameter ExtendField is true, diagonalisation is carried out over the splitting field of its characteristic polynomial.
A new intrinsic IsEtale checks whether an algebra is étale; i.e., diagonalisable over an extension field. If so, a matrix L is returned such that for all elements M of the algebra, LML-1 is diagonal.
The intrinsic CommonEigenspaces returns common eigenvalues and eigenspaces of a sequence Q of commuting matrices. This intrinsic now has parameters U and Check. If U is assigned, only subspaces of U are considered. The default value for U is the base module of the universe of Q. If Check is true (the default value), the matrices in Q are checked for commutativity.
A package for étale algebras developed by Stefano Marseglia has been added. It covers étale algebras that are products of absolute number fields and supports orders and ideals.
New Features:
The computation of noncommutative Gröbner bases for Finitely Presented Associative Algebras has been greatly improved:
The useless pair elimination method of B. Keller (the noncommutative analogue to the Gebauer-Möller criterion) has been implemented. This means that in general, many less critical pairs are needed than previously, so the size of the matrices in the noncommutative F4 algorithm are much smaller.
The Aho-Corasick substring search algorithm has been implemented for words in FP algebras (for a dictionary D of words and given word S, this finds all words in D which are subwords of S). This allows a very fast search for divisors in D of a given noncommutative monomial S, where D are the leading monomials of a partial Gröbner basis.
As a result of both points above, the symbolic reduction phase (which searches for reductors) and the critical pair phase (which does a similar search in the elimination search) have both been very greatly sped up for many kinds of large examples.
New Features:
The Dixon-Schneider algorithm for computing the character table of a finite group has been greatly sped up.
The default algorithm for computing the character table of a finite group now uses the Dixon-Schneider algorithm for groups with small conjugacy classes.
A new fast algorithm for computing the character table of an abelian group has been developed.
Addition of characters (such as returned by GaloisOrbit) has been greatly sped up.
Features:
This package computes group representations for groups that are not necessarily finitely generated, with focus on reductive groups.
Standard constructions such as StandardRepresentation, TrivialRepresentation and
AlternatingRepresentation are available.
For a reductive group, one can also construct a HighestWeightRepresentation, and for orthogonal groups one can construct thr SpinorNormRepresentation, as well as characters induced from the action on the radical via RadicalSignCharacter.