Release Notes V1.30-1 (5 March 1996)

Magma V1.30-1 (5 March 1996)

Here is a terse summary of the new features installed in Magma between November 1, 1994 (the date V1.1 started shipping) and March 5, 1996 (the release date for V1.3).

Change of Semantics

NormalSubgroups no longer synonym for NormalLattice (will now return a sequence of subgroup records)

Packages (New Language feature)

User definitions of intrinsic signatures easily added

Extensive but easy to use syntax to specify particular signatures

Automatic attaching and detaching of package files so that their intrinsics are included into the current Magma intrinsics

Automatic recompilation of packages so that re-load or re-attaching unnecessary

Sharing of constant definitions between package files

Succinct argument checking with 'require', etc.

Specification files for easy collecting together and simultaneous attaching of package files

Automatic attaching of system and user specification files at startup by environment variables

Finitely Generated Abelian Groups (New Category)

Construction as a quotient of a free abelian group

Direct product, free product

Arithmetic

Construction of subgroups and quotient groups

Elementary divisors, primary invariants

Factor basis, divisor basis, primary basis

Torsion subgroup, torsion-free subgroup, p-primary component

Homomorphisms: Image, kernel, cokernel

Composition series, maximal subgroups, subgroup lattice (for a finite group)

Character table (for a finite group)

Abelian quotient of any group (with its natural homomorphism)

Conversion between Z-modules and abelian groups

Functors from rings and fields onto abelian groups

Permutation Groups

The Holt-Leedham-Green-O'Brien random element algorithm

Abelian quotient, soluble quotient

Chief series, chief factors

Elementary abelian series, radical, solvable residual, radical quotient

Socle of a primitive group

An improved algorithm for constructing the O'Nan-Scott decomposition of a primitive group

The performance of the Sylow p-subgroup algorithm has been greatly improved

Normal subgroups

Conjugacy classes of subgroups, poset of subgroup classes

Conjugacy classes of subgroups satisfying a condition: Cyclic, abelian, nilpotent, soluble, simple, perfect, regular

Presentation on given generators (for groups of moderate order)

KG-module corresponding to an elementary abelian section

Irreducible representations (for groups of moderate order)

Matrix Groups

The Holt-Leedham-Green-O'Brien random element generation algorithm

The Murray-O'Brien base selection strategy

Actions on general G-sets

Stabilizer of a subspace

Abelian quotient, soluble quotient

Composition series, composition factors, chief series, chief factors

Characteristic series: Derived, upper central, lower central, elementary abelian, p-central, Jennings

Characteristic subgroups: Centre, derived subgroup, Fitting subgroup, O_p(G), radical, solvable residual

Subgroup constructions: Centralizer, intersection, normal closure, core

Sylow p-subgroup

Conjugacy of elements

Conjugacy classes of elements, class map, power map

Character table

Presentation on given generators (for groups of moderate order)

KG-module corresponding to an elementary abelian section

Molien series

Decomposition of a subgroup of GL(n, q) into primitive components

Decomposition of a subgroup of GL(n, q) as a tensor product

Permutation representation on the cosets of a subgroup

Conversion of a soluble matrix group to a pc-group

Finite Soluble Groups

Permutation representation on the cosets of a subgroup

Composition series

Conjugacy classes of complements of a normal subgroup

Normal subgroups

Conjugacy classes of subgroups, poset of subgroup classes

Conjugacy classes of subgroups satisfying a condition: Cyclic, elementary abelian, abelian, nilpotent

Automorphism group and isomorphism of p-groups (explicit isomorphism returned)

New version of Eamonn O'Brien's p-group generation program

System of double coset representatives for a pair of subgroups

KG-module corresponding to an elementary abelian section

Irreducible representations (for groups of moderate order)

The Rational Field Q and its Ring of Integers Z

Alternative representation of integers in factored form

Faster greatest common divisor algorithm for integers

More efficient arithmetic for Q

Fast SQUFOF factorization for small integers

Multiple polynomial quadratic sieve method of factorization

Cunningham database of factorizations of integers of the form p^n +/- 1

Univariate Polynomial Rings

Karatsuba algorithm for polynomial multiplication

GCD-HEU algorithm for gcd's of polynomials over Z

Modular algorithm for gcd's of polynomials over number fields

Resultant (sub-resultant algorithm, Euclidean algorithm), discriminant

Factorization of polynomials over large prime finite fields (Shoup algorithm)

Factorization of polynomials over Z (Collins-Encarnacion algorithm)

Factorization of polynomials over number fields (Trager algorithm)

Factorization of polynomials over p-adic fields

Construction of ideals and subrings (over K)

Construction of quotient rings

Arithmetic with ideals (over K)

Determination of whether an ideal is: maximal, prime, primary

Radical of an ideal (over K)

Primary decomposition of ideals (over K)

K[x] / I as an algebra over K

Multivariate Polynomial Rings

Monomial orders: lexicographical, graded lexicographical, graded reverse lexicographical, block elimination.

Arithmetic with elements

Recursive coefficient, monomial, term, and degree access

Determination of whether an element is: a unit, a zero-divisor, nilpotent

Differentiation, integration, evaluation and interpolation

Resultant (sub-resultant algorithm), discriminant

Greatest common divisor (sparse EEZ-GCD and fast GCD-HEU algorithms)

Factorization over Z, Q, and rational function fields over Z, Q (EEZ algorithm).

Construction of ideals and subrings

Construction of quotient rings

Groebner bases of ideals

Groebner Walk algorithm for converting the Groebner basis of an ideal with respect to one order to the Groebner basis of the ideal with respect to another order

Arithmetic with ideals (sum, product, intersection, colon ideal, leading monomial ideal)

Normal form of a polynomial with respect to an ideal

S-polynomial of two polynomials

Determination of whether a polynomial is in an ideal or its radical

Determination of whether an ideal is: zero, proper, zero-dimensional

Computation of univariate elimination ideal generators

Computation of the variety of a zero-dimensional ideal

Elimination ideals

Relation ideals (determination of algebraic relations between polynomials)

Syzygy modules

Construction of elementary symmetric polynomials

Determination of whether a polynomial is symmetric, and, if so, expression of it in terms of the elementary symmetric polynomials

Permutation and matrix group actions on polynomials

Valuation Rings

Construction of a valuation ring of a rational function field

Arithmetic

Quadratic Fields and their Orders

Discriminant, signature, conductor

Maximal order, integral basis

Decomposition of primes

Quadratic forms: Reduction, composition, powering, prime form, action of SL(2,Z)

Class number (Shank's algorithm)

Ideal class group (Buchmann's method)

Unit group, fundamental unit, regulator

Factorization of an ideal

Solution of norm equations

General Number Fields and their Orders

Relative extensions

Kummer extensions

Improved ideal arithmetic

Determination of whether an ideal is: integral, prime, principal

Decomposition of primes

Improved algorithms for class group unit group

Solution of relative norm equations and Thue equations

Determination of subfields

Isomorphism of number fields

Determination of Galois groups (for equations of degree less than 12)

The Real and Complex Fields (PARI real package)

Arithmetic

Square root, arithmetic-geometric mean

Continued fraction expansion of a real number

Constants: pi, Euler's constant, Catalan's constant

Logarithm, dilogarithm, exponential

Trigonometric functions and their inverses

Hyperbolic functions and their inverses

Bernoulli numbers

Gamma function, incomplete Gamma function, complementary incomplete Gamma function, logarithm of Gamma function

J-Bessel function, K-Bessel function

U-confluent hypergeometric function

Logarithmic integral, exponential integral

Error function, complementary error function

Dedekind eta function

Jacobi sine theta-function and its k-th derivative

Log derivative (psi) function, i.e, {Gamma'(x)over Gamma(x)}

Riemann-zeta function

Polylogarithm, Zagier's modifications of the polylogarithm

Weber's f-function, Weber's f_2-function

Integer polynomial having a given real or complex number as an approximate root (Hastad, Lagarias and Schnorr LLL-method)

Roots of a polynomial (Schonhage method)

Summation of a series (Euler-Wijngaarden method for alternating series)

Numerical integration of a function (Romberg-type methods)

p-adic Fields (New Category)

Arithmetic operations; valuation of an element

Square root, n-root, logarithm, exponential

Ring of integers of a p-adic field

Factorization of polynomials over a p-adic field

Modules over multivariate polynomial rings

Arithmetic

Row and column operations on elements

Membership test

Construction of submodules

Sum and intersection of submodules

Image, kernel

Syzygy modules

Matrix Algebras

Quotient algebras

Ideals

Sum, intersection, product of ideals

Centre, commutator algebra, Jacobson radical

Maximal (minimal) left, right, two-sided ideals

Elliptic Curves

Creation of an elliptic curve over a field

Models: Weierstrass form, integral model, minimal model

Arithmetic with rational points

Height, local height, height pairing

Invariants: b-invariants, c-invariants, j-invariant, discriminant

Invariants for integral curves: Conductor, regulator, Mordell-Weil rank, Tamagawa numbers

Mordell-Weil group, torsion subgroup

Kodaira symbols

Isomorphism of curves

Enumerative Combinatorics

Factorial

Binomial, multinomial coefficients

Number of partitions of n

Enumeration of restricted and unrestricted partitions

Stirling numbers of the first and second kind

Fibonacci numbers

Graphs

Characteristic polynomial, spectrum

Incidence Structures and Designs (New Category)

Definition of a general incidence structure, near-linear space, linear space, design

Difference sets: Standard difference sets, development

Hadamard designs, Witt designs

Unary operations: complement, contraction, dual, residual

Binary operations: sum, union

Invariants for an incidence structure: point degrees, block degrees, covalence

Invariants for a design: replication number, covalence, intersection numbers, Pascal triangle

Properties: balanced, complete, self-dual, simple

Graphs from designs: block graph, incidence graph, point graph

Automorphism group (J. Leon's algorithm), isomorphism testing

Group actions on a design: orbits and stabilizers of points and blocks

Symmetry properties: point transitive, block transitive

Error-correcting Codes

Trace of a code

Code words of a designated weight

Speed-up of determination of weight distribution/enumerator

Complete weight enumerator

MacWilliams transform for complete weight enumerator

Group actions on a code