Mathematical Areas
Magma covers 45 mathematical areas across 9 major branches of mathematics. Each area is supported by high-performance, research-grade algorithms and comprehensive handbook documentation.
Algebraic Geometry
Varieties and Schemes
Affine and projective varieties, schemes, morphisms, and related constructions over arbitrary fields.
View in Handbook →Coherent Sheaves and Divisors
Coherent sheaves, Weil and Cartier divisors, line bundles, and their cohomology on algebraic varieties.
View in Handbook →Algebraic Curves
Algebraic curves over finite fields, number fields, and function fields, including genus computation, Riemann–Roch, and Jacobians.
View in Handbook →Algebraic Surfaces
Classification of algebraic surfaces, intersection theory, blow-ups, and Picard groups.
View in Handbook →Toric Geometry
Toric varieties from fans and polytopes, Cox rings, and computational tools for toric geometry.
View in Handbook →Associative Algebras
Associative Algebras by Structure Constants
Finite-dimensional associative algebras defined by structure constants, with radical and semisimple decomposition.
View in Handbook →Finitely Presented Algebras
Algebras presented by generators and relations with Gröbner basis methods for rewriting and ideal membership.
View in Handbook →Matrix Algebras
Algebras of matrices over fields and rings, including decomposition, centraliser algebras, and lattice methods.
View in Handbook →Quaternion Algebras
Quaternion algebras over number fields and local fields, maximal orders, and arithmetic of quaternionic lattices.
View in Handbook →Combinatorial Theory
Introduction to Combinatorial Theory
Combinatorial structures in Magma — an overview of the major areas including graphs, designs, and codes.
View in Handbook →Partitions, Young Tableaux and Symmetric Functions
Integer partitions, standard and semistandard Young tableaux, and symmetric function algebra.
View in Handbook →Graph Theory
Construction and analysis of graphs and digraphs, automorphism groups, colouring, planarity, and graph databases.
View in Handbook →Design Theory
Combinatorial designs — BIBDs, resolutions, and automorphism groups, with extensive databases.
View in Handbook →Finite Planes
Projective and affine planes, collineation groups, and constructions from difference sets.
View in Handbook →Hadamard Matrices
Hadamard matrices and their equivalences, automorphism groups, and constructions.
View in Handbook →Commutative Algebra
Gröbner Bases
Buchberger's algorithm and the Faugère F4/F5 variants for computing Gröbner bases over fields and rings.
View in Handbook →Multivariate Polynomial Rings and Their Ideals
Ideals in multivariate polynomial rings — primary decomposition, radical, saturation, and elimination.
View in Handbook →Invariant Theory
Rings of invariants for linear group actions via Gröbner bases and Molien series.
View in Handbook →Modules over Polynomial Rings
Free resolutions, Hilbert functions, syzygies, and Ext/Tor for modules over polynomial rings.
View in Handbook →Commutative Rings
Introduction to Commutative Rings
An overview of the commutative ring hierarchy in Magma — from integers through to general rings.
View in Handbook →Integer Rings
Arithmetic in integer rings including GCDs, factorisation, primality tests, and number-theoretic functions.
View in Handbook →Finite Fields
Finite fields of prime and prime-power order, field extensions, Frobenius, and discrete logarithms.
View in Handbook →Real and Complex Fields
Real and complex number fields with arbitrary precision arithmetic and certified root-finding.
View in Handbook →Polynomial Rings
Univariate and multivariate polynomial rings — factorisation, GCDs, resultants, and discriminants.
View in Handbook →General Commutative Rings
Localisation, quotient rings, tensor products, and general constructions for commutative rings.
View in Handbook →Error-Correcting Codes
Linear Codes over Finite Fields
Generator matrices, minimum distance, decoding algorithms, and standard families of linear codes.
View in Handbook →Linear Codes over Finite Rings
Linear codes over rings including Z/4Z, with Gray maps and connections to lattices.
View in Handbook →Additive Codes
Additive codes over fields, including stabiliser codes and associated combinatorial structures.
View in Handbook →Quantum Error-correcting Codes
Quantum stabiliser codes, CSS construction, and connections to classical linear codes.
View in Handbook →Group Theory
Finitely Presented Groups
Coset enumeration, Reidemeister–Schreier, abelianisation, and low-index subgroups of finitely presented groups.
View in Handbook →Permutation Groups
Base and strong generating sets, conjugacy classes, normal structure, maximal subgroups, and automorphism groups of permutation groups.
View in Handbook →Matrix Groups
Matrix groups over finite fields and number fields, composition tree, natural representations, and constructive recognition.
View in Handbook →Abelian Groups
Finitely generated abelian groups, structure theorem, Smith normal form, and the abelian group database.
View in Handbook →Modules and Representation Theory
Module Theory
Modules over rings and algebras, submodule lattices, composition series, and homomorphism spaces.
View in Handbook →Representation Theory
Ordinary and modular representations of finite groups — characters, Brauer characters, and decomposition matrices.
View in Handbook →Number Theory
Elementary Number Theory
Divisibility, Euler's function, primitive roots, quadratic residues, and continued fractions.
View in Handbook →Algebraic Number Theory
Number fields, rings of integers, ideal class groups, unit groups, and class field theory.
View in Handbook →Local Rings and Fields
p-adic fields, unramified and totally ramified extensions, and local arithmetic.
View in Handbook →Lattices
Lattice reduction (LLL, BKZ), shortest vector, sphere packing, and Gram matrices over number fields.
View in Handbook →Elliptic Curves over Q and Number Fields
Mordell–Weil group, descent, L-functions, isogenies, and the Birch–Swinnerton-Dyer conjecture for elliptic curves.
View in Handbook →Hyperelliptic Curves over Q and Number Fields
Jacobians, 2-descent, p-adic height pairings, and arithmetic of hyperelliptic curves.
View in Handbook →Curves over Finite Fields
Point counting (Schoof, Harvey–Sutherland), zeta functions, and isogeny classes for curves over finite fields.
View in Handbook →Artin Representations
Artin representations, conductor, Artin L-functions, and connections to Galois theory.
View in Handbook →L-Functions
Numerical computation of L-functions, zeros, special values, and connections to modular forms.
View in Handbook →Modular Forms
Spaces of modular forms and cusp forms, Hecke operators, newforms, and associated L-functions.
View in Handbook →