Release Notes V2.4 (December 14, 1998)

This screen provides only a short summary of the new features installed in Magma for release version V2.4. For a more detailed listing of the new features including information about changes that arise with this version click here.

Summary

Two important group theory databases are now included: the Besche-Eick database of all groups up to order 1000 (excepting orders 512 and 768) and the Hulpke database of transitive groups of degree up to 22.

A facility for analyzing matrix groups of large degree defined over finite fields has been installed. This module uses the Aschbacher classification of maximal subgroups of GL(n, q). The module was written by Derek Holt, Alice Niemeyer, Eamonn O'Brien and Anthony Pye.

The Schoenhage-Strassen FFT-based algorithm for the multiplication of very large integers and an asymptotically-fast algorithm for the division of large integers has been implemented.

Two separate FFT-based algorithms are now employed for the multiplication of univariate polynomials over the integers and similar coefficient rings. Also, an asymptotically-fast algorithm for the division of polynomials has been implemented and modular exponentiation is now performed using pre-inversion of the modulus.

A new efficient packed representation for polynomials over GF(2) has been implemented. This has led to dramatic speed-ups for all computations within finite fields of characteristic 2. A database of sparse irreducible polynomials over GF(2) has also been constructed for all degrees up to 11000.

The Shoup algorithm for factorization of polynomials over finite fields has been implemented, leading to very significant speed-ups.

Stage 1 of a major revision and extension of the power series module has been completed. In this stage, the existing machinery has been redone in greater generality and asymptotically-fast algorithms are employed for multiplication and division. Series with fractional exponents are now permitted.

The KANT number field machinery corresponding to Kash 1.9 has been installed. This is the first major upgrade of the Magma general number field facility since Kash V1.5 in 1995. This version of KANT provides the Magma user with greatly enhanced performance for many fundamental algorithms. Particularly noteworthy is a Round 4 integral basis algorithm and a much improved class group algorithm.

Major new number field facilities available in Magma V2.4 (courtesy of KANT) are ray class groups, S-unit groups, prime ideal decompositions of fractional ideals, subfield lattices, automorphism groups of normal and abelian extensions, solution of Thue equations, unit equations and index form equations.

A set of functions have been provided for computing q-expansions of the standard elliptic and modular functions. These include the Weierstrass p-function, Eisenstein series, Dedekind eta-function, Jacobi theta-function, elliptic j-invariant and discriminant function.

The facilities for elliptic curves have undergone major expansion. Noteworthy is the introduction of general machinery for working with isomorphisms, isogenies and rational maps between curves. The new machinery allows comparatively easy computation of things such as the endomorphism ring of a curve over a finite field and the eigenvalues of the Frobenius automorphism (the latter being useful in point-counting).

The Cremona database of elliptic curves having conductor up to 5300 has been installed.

The design of the facility for finite planes has been revised, firstly, to make it easier to work with planes and their subplanes and secondly, to make it follow similar conventions to other structures.

Details

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