Intrinsics

An intrinsic is a builtin Magma function or procedure. Most intrinsics perform some standard algebraic operation, though a few relate to the Magma system itself. A good way of getting information about an intrinsic is to use this help system. For summary information, you may obtain the intrinsic's signature. See Signature.

The name of almost every intrinsic starts with a capital letter. If the common mathematical name for what the intrinsic does has more than one word, then the Magma name will probably be the same name, with capital letters starting each word and with no intervening spaces. For instance, the intrinsic for finding the primitive root is called PrimitiveRoot. Several intrinsics have two names: a long explanatory one and a short convenient one. For instance, AbelianQuotientInvariants and AQInvariants are the same.

Almost all intrinsics have arguments. For instance, HammingCode(K, r) takes two arguments, a finite field K and an integer r. You could use it as

> HC := HammingCode(FiniteField(2), 4);

Some intrinsics have parameters. These are optional values supplied after the arguments and a colon (:) . If you do not specify a parameter, then Magma will use its "default value". For instance, the intrinsic IsSubsequence(Q, R) has a parameter called Kind, with a default value of "Consecutive":

> S1 := [6, 5];                                      
> S2 := [2, 6, 3, 5, 7];
> print IsSubsequence(S1, S2);                       
false
> print IsSubsequence(S1, S2 : Kind := "Sequential");
true

If a Boolean parameter B of an intrinsic I is being given the value true, then the syntax

I(ARGS : PARAM := EXPR, PARAM := EXPR, ..., B)

I(ARGS : PARAM := EXPR, PARAM := EXPR, ..., B := true)

The library of intrinsic functions in Magma contains implementations of approximately 2000 algebraic and geometric algorithms. Highlights include:

• Fast machinery for polynomial algebra, particularly for the factorization of univariate and multivariate polynomials over Z and several kinds of fields. Magma supports calculation with ideals in rings. Groebner basis algorithms are employed.
• An extensive modules for finite fields that provides users with the capability of computing in fields as large as GF(2^1000). Various optimal representations of Galois fields are used to provide superior performance over a wide range of cardinalities. The module also solves the compatibility problem for fields constructed by different chains of extensions.
• The KANT system developed by Michael Pohst's group for constructive algebraic number theory. This package includes facilities for calculating integral bases, ideal class groups and systems of fundamental units for fields of degree in excess of 20.
• The Pari system developed by Henri Cohen and collaborators in Bordeaux is included as part of Magma. It provides users with access via the Magma programming language to the highly efficient Pari code for elliptic curves, number fields, real and complex field, and power and Laurent series.
• A new generation of permutation group algorithms that enable users routinely to perform computations, such as computing a chief series, in short base groups having degree up to at least a million. Backtrack searches are conducted using Jeff Leon's new package (PERM).
• New algorithms and programs for most of the critical processes for finitely presented groups. Included among these are George Havas's new Todd-Coxeter procedure, Eamonn O'Brien's p-quotient and p-group generation programs and Sims' new backtrack-based low index subgroups algorithm.
• Extensive machinery for computing with modules and their homomorphisms. This includes highly optimized code for modules over finite fields and over the ring of integers.
• A graph theory module containing many functions for determining graph invariants as well as Brendan McKay's graph automorphism program (nauty).
• A coding theory package including many constructions for linear codes together with machinery for determining important invariants such as the minimum weight, the weight enumerator and the automorphism group.