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
  • Subgroups of PSL(2,R)
• 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
  • 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
  • Galois Rings
  • Number Fields and their Orders
    • Number Fields
    • Orders and Fractional Ideals
    • Invariants
    • Diophantine Equations
    • Automorphisms
    • Class Field Theory
    • Quadratic Fields
    • Cyclotomic 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 of Global Algebraic Function Fields
    • Class Field Theory for Algebraic Function Fields
  • Discrete Valuation Rings
  • The Real and Complex Fields
  • Newton Polygons
  • Local Rings and Fields
    • Local Rings: Construction
    • Local Rings: Arithmetic
    • Local Rings: Polynomial Factorization
  • Power, Laurent and Puiseux Series Rings
  • Lazy Power Series Rings
  • Algebraically Closed Fields
• Commutative Algebra
  • Multivariate Polynomial Rings
    • Polynomial Rings: Creation and Ring Operations
    • Polynomial Rings: Arithmetic with Polynomials
    • Polynomial Rings: GCD and Factorization
    • Polynomial Rings: Gröbner Basis
    • Polynomial Rings: Arithmetic with Ideals
    • Polynomial Rings: Invariants for Ideals
    • Polynomial Rings: Gradings
  • Affine Algebras
    • Affine Algebras: Creation and Operations
    • Affine Algebras: Arithmetic with Ideals
  • Modules over Affine Algebras
    • Modules over Affine Algebras: Creation and Operations
    • Modules over Affine Algebras: Submodules
    • Modules over Affine Algebras: Homology
• Linear Algebra and Module Theory
  • Matrices
    • Representation of Matrices
    • Arithmetic
    • Echelon Form and Nullspace
    • Canonical Forms
  • Sparse Matrices
  • Vector Spaces
    • Construction
    • Construction
    • Subspaces and Quotient Spaces
    • Bases
    • Homomorphisms
    • Quadratic Forms
  • Free Modules
    • Basic Operations
    • Homomorpisms
  • Modules over Dedekind domains
• Lattices and Quadratic Forms
  • Lattices
    • Lattices: Construction and Operations
    • Lattices: Properties
    • Lattices: Reduction
    • Lattices: Automorphisms
    • Lattices: Neighbors and Genera
    • Lattices: G-Lattices
  • Binary Quadratic Forms
• Algebras
  • Finitely Presented Associative Algebras
  • General Finite-Dimensional Algebras
  • Finite-Dimensional Associative Algebras
    • Orders of Associative Algebras
  • Quaternion Algebras
  • Group Algebras
  • Matrix Algebras
  • Finite-Dimensional Lie Algebras
    • Lie Algebras: Construction and Arithmetic
    • Lie Algebras: Properties and Invariants
    • Lie Algebras: Arithmetic of Subalgebras and Ideals
    • Lie Algebras: Structure
    • Lie Algebras: Representations
    • Lie Algebras: universal enveloping algebras
  • Quantized Enveloping Algebras
• Representation Theory
  • Modules over an Algebra
    • A-Modules: Creation
    • A-Modules: Constructions
    • A-Modules: Submodules and Quotient Modules
    • A-Modules: Structure
    • A-Modules: Homomorphisms
  • Representations of Symmetric Groups
  • Character Theory
  • Invariants of Finite Groups
    • Construction of Primary and Secondary Invariants
    • The Ring of Invariants
    • Properties
    • Invariants of the Symmetric Group
• Homological Algebra
  • Basic Algebras
  • Chain Complexes
• Lie Theory
  • Coxeter systems
  • Root Systems
    • Creating Root systems
    • Operations and properties for Root Data
    • Roots and coroots
    • New root systems from old
  • Root data
    • Creating Root data
    • Operations and properties for Root Data
    • Roots, coroots and weights
    • New root systems from old
    • Constants
  • Coxeter Groups
  • Coxeter Groups as Permutation Groups
  • Complex Reflection Groups
  • Groups of Lie Type
    • Creating Groups of Lie type
    • Operations and Properties
    • Automorphisms
    • Representation Theory
• Algebraic Geometry
  • Schemes
    • Schemes: Ambient Spaces
    • Schemes: Creation and Properties
    • Schemes: Mappings
    • Schemes: Automorphisms
    • Schemes: Isomorphic Projections
    • Schemes: Linear Systems
  • General Algebraic Curves
    • Curves: Construction and General Properties
    • Curves: Mappings
    • Curves: Local Analysis
    • Curves: Function Field
    • Curves: Divisors and the Riemann-Roch Theorem
    • Curves: Differentials
  • Rational Curves and Conics
  • Elliptic Curves
    • Elliptic Curves: Construction and Properties
    • Elliptic Curves: Morphisms
    • Elliptic Curves: Operations over
    • Elliptic Curves: Operations over Algebraic Number Fields
    • Elliptic Curves: Operations over F_q
  • Hyperelliptic Curves
    • Hyperelliptic Curves: Construction and Properties
    • Hyperelliptic Curves: Morphisms
    • Hyperelliptic Curves: Operations over F_q
    • Hyperelliptic Curves: Operations over Number Fields
    • Hyperelliptic Curves: Construction of the Jacobian
    • Hyperelliptic Curves: Operations on the Jacobian over
    • Hyperelliptic Curves: Operations on the Jacobian over F_q
    • Hyperelliptic Curves: Kummer Surface
  • Modular Forms
    • Module of Supersingular Points
    • Modular Forms
    • Modular Symbols
    • Brandt Modules
    • Dirichlet Characters
    • Classical Modular Forms and Functions
    • Modular Curves
    • Modular Abelian Varieties
  • Graded Rings and Geometric Databases
  • Resolution Graphs and Splice Diagrams
• Differential Galois Theory
  • Differential Rings and Fields
  • Differential Operator Rings
• Finite Incidence Structures
  • Enumerative Combinatorics
  • Partitions and Young Tableaux
  • Symmetric Function Algebras
  • Graphs
  • Networks
  • Incidence Structures and Designs
  • Finite Planes
  • Incidence Geometry
    • Incidence Geometries
    • Coset Geometries
• Error-correcting Codes
  • Linear Codes over Finite Fields
    • Linear Codes: Creation
    • Linear Codes: Operations on Codewords
    • Linear Codes: Elementary Operations
    • Linear Codes over Finite Fields: Basic Properties
    • Linear Codes over Finite Fields: Weight Distribution
    • Linear Codes over Finite Fields: Construction of Families
    • Linear Codes over Finite Fields: Algebraic-Geometric codes
    • Linear Codes over Finite Fields: Decoding Algebraic-Geometric codes
    • Linear Codes over Finite Fields: Changing the Alphabet
    • Linear Codes over Finite Fields: Combining Codes
    • Linear Codes over Finite Fields: Bounds
    • Linear Codes over Finite Fields: Best Known Codes
    • Linear Codes over Finite Fields: Decoding Algorithms
    • Linear Codes over Finite Fields: Automorphism Group
    • Linear Codes over Finite Fields: LDPC Codes
    • Linear Codes over Finite Fields: Attacks on the McEliece Cryptosystem
  • Linear Codes over Finite Rings
    • Linear Codes over Finite Rings: Creation
    • Linear Codes over Finite Rings: Operations on Codewords
    • Linear Codes over Finite Rings: Elementary Operations
    • Linear Codes over Finite Rings: Weight Distribution
    • Linear Codes over
    • Linear Codes over
    • Linear Codes over
  • Additive Codes
    • Additive Codes: Creation
    • Additive Codes: Operations on Codewords
    • Additive Codes: Elementary Operations
    • Additive Codes over Finite Fields: Basic Properties
    • Additive Codes over Finite Fields: Weight Distribution
    • Additive Codes over Finite Fields: Construction of Families
    • Additive Codes over Finite Fields: Automorphism Group
  • Quantum Error-Correcting Codes
    • Quantum Codes: Creation
    • Quantum Codes: Basic Properties
    • Quantum Codes: Error Group
    • Quantum Codes: Weight Distribution
    • Quantum Codes: Constructions
    • Quantum Codes: Automorphism Group
• Cryptography
  • Pseudo-Random Sequences
• Mathematical Databases
  • Group Theory
  • Number Theory
  • Algebraic Geometry
  • Topology
  • Incidence Structures
  • Linear Codes
• Documentation
• References