Magma includes a growing number of mathematical databases. Typically, such a database contains a complete classification of all structures of some given type up to a specified bound. A number of these databases are an integral part of algorithms installed in Magma. The current databases include:
Small Groups: The Small Groups Library developed by Besche, Eick and O'Brien. This database contains all groups of order up to 2000, except the groups of order 1024, and a number of infinite series of larger groups.
The ATLAS Database: Representations of nearly simple groups, as in the Birmingham ATLAS of Finite Group Representations. The data was supplied by Rob Wilson.
Almost Simple Groups: A database of almost simple groups stored with their automorphism groups and maximal subgroups. In particular, the database includes all simple groups having a permutation representation of degree less than 1000. The following groups are included:
Alternating groups: An for n ≤ 999;
Classical groups: L2(q), L3(q), L4(q) and L5(q) for all prime powers q; L6(3), L7(3); Ld(2) for d ≤ 14;
Sp4(q) for all odd prime powers q and even q ≤ 16; Sp6(3), Sp8(2), Sp10(2);
U3(q) for all primes q and prime powers q ≤ 25; U4(q) for prime powers q ≤ 7, U6(2);
O7(3), O±8(2), Om8(3), O±10(2)
Exceptional groups: G2(3), G2(4), G2(5)
Twisted groups: Sz(8), Sz(32), 3D4(2), 2F4(2)'
Sporadic groups: M11, M12, M22, M23, M24, J1, J2, J3, HS, McL, He, Co2, Co3, Fi22
Simple Groups: A database containing a presentation, the conjugacy classes and maximal subgroups for each simple group of order less than a million. The database was prepared by Jamali, Robertson and Campbell.
Perfect Groups : The database of perfect groups of order up to a million constructed by Holt and Plesken.
Transitive Groups: The transitive permutation groups of degree up to 30. The transitive groups of degree up to 15 were determined by Butler and McKay while the classification was extended to degree 30 by Hulpke.
Primitive Groups: The table of primitive groups of degree up to 2499 (Sims, Roney-Dougal &Unger, Roney-Dougal).
Permutation Representations: A collection of finite groups given in terms of permutation representations. A particular group is included if:
It is an 'interesting' group. In practice this means a sporadic simple group or a close relative of such; or
It is representative of some class of groups which is useful for testing conjectures and algorithms.
Irreducible Matrix Groups: The table of irreducible subgroups of GL(n,p) where p is prime and pn ≤ 2499 (Sims, Roney-Dougal &Unger, Roney-Dougal).
Irreducible Soluble Groups: The irreducible soluble subgroups of GL(n, p) for n > 1 and pn < 256, as classified by Short.
Finite Groups of Rational Matrices: The maximal finite subgroups of GL(n,ℚ), for n up to 31.
Quaternionic Matrix Groups: The finite absolutely irreducible subgroups of GLn(D) where D is a definite quaternion algebra whose centre has degree d over ℚ and nd ≤ 10
Matrix Representations: A collection of modular representations of simple groups (mainly sporadic groups) and coverings of simple groups. This collection is a subset of the Parker database of modular representations.
Cunningham Factorizations: A database containing 237,578 factors f of numbers an±1, where a < 10000,n < 10000, and f > 109. The factorizations of integers of the form an±1, a ≤ 12, were produced by Sam Wagstaff and collaborators (for n up to 1200), with contributions from Arjen Bot, Will Edgington, Alexander Kruppa and Paul Leyland (for larger n); for 13 < a < 99 they are mainly from the Brent-Montgomery-te Riele extension of the Cunningham tables, with contributions by ECMNET and various individuals; for 100 < a < 1000 they are mainly from the tables produced by Hisanori Mishima and Mitsuo Morimoto with additions from Rob Hooft, Pete Moore and others. For prime bases a < 1000 Richard Brent computed many of the factors for an unpublished extension of the Brent-Montgomery-te Riele tables.
Irreducible polynomials: A database of sparse irreducible polynomials over GF(2) for all degrees up to 23,030, constructed by Allan Steel. The polynomials have the form f(x) = xn + g(x), where g(x) is of minimal degree such that f(x) is irreducible.
Conway polynomials: A database of Conway polynomials for F2 to F127. These polynomials are primitive and provide standard definitions for finite field extensions (used in modular representation theory, for example). This database is based on lists built by Richard Parker and Frank Lübeck.
Galois Polynomials: For each transitive group G with degree between 2 and 15, the database contains a univariate polynomial over the integers which has G as its Galois group. These polynomials have been determined by J. Klüners and G. Malle.
Cremona database of Elliptic Curves: A database constructed by John Cremona that contains all isogeny classes of elliptic curves having conductor up to 30,000 is available.
Stein-Watkins Database of Elliptic Curves: The Stein-Watkins database of 136,924,520 elliptic curves of conductor up to 108 is now available in Magma.
K3 Surfaces: This comprises a collection of 24,099 K3 surfaces. For g = -1,0,1,2, all K3 surfaces of genus g are included, there being 4281, 6479, 6627 and 6628 surfaces, respectively. For higher genus, the data associated to the 6628 K3 surfaces of genus 2 propagates in a predictable way, so only those K3 surfaces with codimension at most 7 and genus in the range 3 to 9 are included.
3-folds: Basic machinery is provided that allows the user to generate lists of Fano 3-folds and Calabi–Yau 3-folds.
Fundamental Groups of 3-manifolds: A database containing the 11,126 small-volume closed hyperbolic manifolds of the Hodgson-Weeks census. Each manifold record contains a presentation of the fundamental group and a homomorphism to Sn whose kernel has positive betti number.
Simple Graphs: Magma contains an interface to the graph enumeration program of Brendan McKay which allows the user to rapidly construct all simple graphs on a given number of vertices. The graph generation program allows the user to specify one or more conditions thereby allowing the construction of all graphs on a given number of vertices satisfying the condition.
Strongly Regular Graphs: A database containing a list of strongly regular graphs constructed by Brendan McKay, Ted Spence and others. This database contains strongly regular graphs on 25, 26, 27, 28, 29, 35, 36, 37 and 40 vertices. The graphs are indexed by the order of the graph, its degree, the number of common neighbours to each pair of adjacent vertices, and the number of common neighbours to each pair of non-adjacent vertices.
Hadamard Matrices; This database includes all inequivalent Hadamard matrices of degree at most 28, and examples of matrices of all degrees up to 256. The matrices up to degree 28 have been provided by Neil Sloane while most of those of larger order have been provided by C. Koukouvinos, I. Kotsireas and G. Stelios.
Skew Hadamard Matrices; This database includes known skew Hadamard matrices of degree up to 256. The matrices up to degree 28 have been provided by Neil Sloane while most of those of larger order have been provided by C. Koukouvinos, I. Kotsireas and G. Stelios.
Best Known Binary Linear Codes: A database containing constructions of the best known linear codes over F2 of length up to 256 has been implemented by M. Grassl and the Magma group from tables of A. E. Brouwer. The codes of length up to 36 are optimal. The database is complete in the sense that it contains a construction for every set of parameters. Thus the user has access to 33,152 binary codes.
Best Known F3 Linear Codes: A database containing constructions of the best known linear codes over F3 of length up to 100. This database has been constructed by M. Grassl and the Magma group.
Best Known F4 Linear Codes: A database containing constructions of the best known linear codes over F4 of length up to 100. This database has been constructed by M. Grassl and the Magma group.