Contents

• Introduction
  • The Magma Philosophy
  • Summary of this Document
• The Magma Language and System
  • The Magma User Language
  • The Magma Environment
• Groups
  • Permutation Groups
    • Construction
    • Base and Strong Generating Set
    • Elementary Properties
    • Conjugacy Classes
    • Subgroup Constructions
    • Actions
    • Analysis of a Primitive Group
    • Normal Structure
    • Standard Quotients
    • Subgroup Structure
    • Automorphisms
    • Cohomology and Representations
    • Databases
  • Matrix Groups
    • Construction
    • Arithmetic with Elements
    • Actions
    • Base and Strong Generating Set
    • Elementary Properties
    • Conjugacy Classes
    • Subgroup Constructions
    • Normal Structure
    • Standard Quotients
    • Automorphisms
    • Cohomology and Representations
    • Aschbacher Analysis
    • Databases of Matrix Groups
  • Constructive Recognition
  • Finitely Presented Groups
    • Free Groups
    • Construction
    • Arithmetic on Elements
    • Basic Properties
    • Quotients
    • Constructing a Subgroup
    • Operations on Subgroups of Finite Index
    • Enumeration of Subgroups
    • Simplifying a Presentation
    • Actions
    • Representation Theory
    • Databases of Finitely Presented Groups
  • Generic Abelian Groups
  • Finitely-Presented Abelian Groups
  • Polycyclic Groups
    • Polycyclic Groups: Construction and Arithmetic
    • Polycyclic Groups: Basic Invariants
    • Polycyclic Groups: Subgroup Constructions
    • Polycyclic Groups: Normal Structure
  • Finite Soluble Groups
    • Construction
    • Conjugacy Classes
    • Subgroup Constructions
    • Normal Structure
    • Subgroup Structure
    • Automorphisms and Representations
  • Finite p-Groups
    • Construction
    • Normal Structure
    • Isomorphisms and Automorphism Groups
    • Character Theory
  • Groups Defined by Rewrite Systems
  • Automatic Groups
  • Groups with Elements given as Straight-Line Programs
  • Braid Groups
    • Constructing and Accessing Braid Groups
    • Constructing and Accessing Elements
    • Normal Forms of Elements
    • Arithmetic Operations with Elements
    • Boolean Predicates
    • Lattice Operations
    • Conjugates
    • Homomorphisms
• Semigroups and Monoids
  • Finitely Presented Semigroups
  • Monoids Defined by Rewrite Systems
• Rings and their Fields
  • The Rational Field and its Ring of Integers
    • Arithmetic
    • Residue Class Rings of Z
    • Primality and Factorization
    • The Number Field Sieve
  • Dirichlet Characters
  • Univariate Polynomial Rings
    • Creation and Ring Operations
    • Creation of Special Polynomials
    • Arithmetic with Polynomials
    • GCD and Factorization
    • Arithmetic with Ideals
  • Residue Class Rings of Univariate Polynomial Rings
  • Finite Fields
    • Construction
    • Arithmetic
    • Roots and Polynomial Factorization
    • Discrete Logarithms
    • Derived Structures
• Global Arithmetic Fields
  • Number Fields and their Orders
    • Number Fields
    • Orders and Fractional Ideals
    • Class Groups and Units
    • Diophantine (and other) Equations
    • Quadratic Fields
    • Cyclotomic Fields
  • Galois Theory of Number Fields
  • Class Field Theory of Number Fields
  • General Algebraic Function Fields
    • Rational Function Fields
    • Algebraic Function Fields
    • Orders of Algebraic Function Fields
    • Elements of Algebraic Function Fields and their Orders
    • Ideals of Orders of Algebraic Function Fields
    • Places of Algebraic Function Fields
    • Divisors of Algebraic Function Fields
    • Differentials of Algebraic Function Fields
    • Divisor Class Groups for Global Algebraic Function Fields
    • Class Field Theory for Algebraic Function Fields
  • Algebraically Closed Fields
• Local Arithmetic Fields
  • Discrete Valuation Rings
  • The Real and Complex Fields
  • Newton Polygons
  • p-adic Rings and their Extensions
    • Construction
    • Arithmetic
    • Polynomial Factorization
    • Class field theory
  • General Local Fields
  • Galois Rings
  • Power, Laurent and Puiseux Series Rings
  • Lazy Power Series Rings
• Linear Algebra and Module Theory
  • Matrices
    • Representation of Matrices
    • Arithmetic
    • Echelon Form and Related Operations
    • Canonical Forms
  • Sparse Matrices
  • Vector Spaces
    • Construction
    • Construction
    • Subspaces and Quotient Spaces
    • Bases
    • Homomorphisms
    • Quadratic Forms
  • Free Modules
    • Basic Operations
    • Homomorphisms
  • Modules over Dedekind domains
  • Chain Complexes
• Lattices and Quadratic Forms
  • Integral Lattices
    • Constructions and Operations
    • Reduction Algorithms
    • Properties
  • Lattices with Group Action: G-Lattices
    • Automorphism Groups
    • G-Lattices
  • Quadratic Forms
    • General Quadratic Forms
    • Binary Quadratic Forms
• Associative Algebras
  • Finitely Presented Associative Algebras
  • Exterior Algebras
  • General Finite-Dimensional Algebras
  • Finite-Dimensional Associative Algebras
    • Constructions and Element Operations
    • Ideal and Subalgebra Structure
    • Orders
  • Matrix Algebras
    • Constructions and Element Operations
    • Ideal and Subalgebra Structure
    • Presentations
  • Group Algebras
  • Quaternion Algebras
  • Basic Algebras
    • Construction
    • Modules and Cohomology
• Representation Theory
  • Modules over an Algebra
    • Creation
    • Constructions
    • Submodules and Quotient Modules
    • Structure
    • Homomorphisms
  • Ordinary Representations
    • Splitting of G-modules
    • Construction of Irreducible G-modules
    • Change of Ring
    • Properties
  • Representations of Symmetric Groups
  • Character Theory
• Lie Theory
  • Coxeter systems
  • Root Systems
    • Creating Root Systems
    • Operations and Properties
    • Roots and Coroots
  • Root Data
    • Constructions
    • Operations and Properties
    • Roots, Coroots and Weights
  • Coxeter Groups
    • General Coxeter Groups as FP-Groups
    • Finite Coxeter Groups as Permutation Groups
  • Complex Reflection Groups
  • Finite-Dimensional Lie Algebras
    • Construction and Arithmetic
    • Properties and Invariants
    • Arithmetic of Subalgebras and Ideals
    • Structure
    • Representations
    • Universal enveloping algebras
    • Finitely Presented Lie Algebras
  • Quantized Enveloping Algebras
  • Groups of Lie Type
    • Creating Groups of Lie type
    • Operations and Properties
    • Automorphisms
    • Representation Theory
  • Finite Groups of Lie Type
• Commutative Algebra
  • Multivariate Polynomial Rings
    • Creation and Ring Operations
    • Arithmetic
    • GCD and Factorization
    • Gröbner Basis
    • Arithmetic with Ideals
    • Invariants for Ideals
    • Gradings
  • Boolean Polynomial Rings
  • Affine Algebras
    • Creation and Operations
    • Arithmetic with Ideals
  • Modules over Multivariate Rings
    • Creation and Operations
    • Submodules
    • Homology
  • Invariants for Groups
    • Construction of Primary and Secondary Invariants
    • The Ring of Invariants
    • Properties
    • Invariants of Linear Algebraic Groups
    • Invariant Fields
    • Invariants of the Symmetric Group
• Algebraic Geometry
  • Schemes
    • Ambient Spaces
    • Creation and Properties
    • Mappings
    • Automorphisms
    • Isomorphic Projections
    • Linear Systems
  • Coherent Sheaves
  • General Algebraic Curves
    • Construction and General Properties
    • Mappings
    • Automorphism groups
    • Local Analysis
    • Ordinary Plane Curves
    • Function Field
    • Divisors and the Riemann-Roch Theorem
    • Differentials
    • Resolution Graphs and Splice Diagrams
  • Algebraic Surfaces
    • Formal Desingularisation and Classification
    • Parametrization Of Rational Surfaces
  • Toric Varieties
    • Toric lattices
    • Cones and Polyhedra
    • Fans
    • Basic operations on toric varieties
    • Maps of toric varieties
    • Invariant divisors and Riemann-Roch spaces
    • Mori theory
  • Graded Rings and Geometric Databases
• Arithmetic Geometry
  • Rational Curves and Conics
  • Elliptic Curves
    • Construction and Properties
    • Morphisms
    • Features over finite fields
    • Features over Q
    • Rational points over Q
    • Features over number fields
    • Rational points over number fields
    • Features over function fields
    • Rational points over function fields
  • Models of Genus One Curves
  • Hyperelliptic Curves and Jacobians
    • Construction and Properties
    • Operations on curves over finite fields
    • Operations on curves over Q
    • Construction of Jacobians
    • Kummer Surface of a genus 2 Jacobian
    • Operations on Jacobians over finite fields
    • Operations on Jacobians over Q
  • L-Series
• Modular Arithmetic Geometry
  • Modular Curves
  • Congruence Subgroups of PSL(2,R)
  • Modular Forms
  • Modular Symbols
  • Modular Abelian Varieties
  • Brandt Modules
  • Supersingular Divisors on Modular Curves
  • Hilbert Modular Forms
  • Modular Forms on Imaginary Quadratic Fields
  • Admissible Representations of GL₂(Q_p)
• Differential Galois Theory
  • Differential Rings and Fields
  • Differential Operator Rings
• Geometry
  • Finite Planes
  • Incidence Geometry
    • Incidence Geometries
    • Coset Geometries
  • Cones and Polyhedra
• Combinatorial Theory
  • Enumerative Combinatorics
  • Partitions and Young Tableaux
  • Symmetric Function Algebras
  • Graphs
  • Multigraphs
  • Networks
  • Incidence Structures and Designs
  • Hadamard Matrices
• Coding Theory
  • Linear Codes over Finite Fields
    • Linear Codes: Creation
    • Operations on Codewords
    • Elementary Operations
    • Basic Properties
    • Weight Distribution
    • Construction of Families
    • Algebraic-Geometric codes
    • Decoding Algebraic-Geometric codes
    • Changing the Alphabet
    • Combining Codes
    • Bounds
    • Best Known Codes
    • Decoding Algorithms
    • Automorphism Group
    • Attacks on the McEliece Cryptosystem
  • Low-Density Parity-Check Codes
  • Linear Codes over Finite Rings
    • Constructions
    • Elementary Operations and Properties
    • Weight Distribution
  • Additive Codes
    • Constructions
    • Operations and Properties
    • Weight Distribution
  • Quantum Error-Correcting Codes
    • Constructions
    • Quantum Codes: Basic Properties
• Cryptography
  • Pseudo-Random Sequences
• Mathematical Databases
  • Algebraic Geometry
  • Coding Theory
  • Finite Fields
  • Graph Theory
  • Group Theory
  • Hadamard Matrices
  • Lattices
  • Lie Algebras
  • Number Theory
  • Topology
• Documentation
• References