22 Mathematical Databases

Cremona Database of Elliptic Curves: A database constructed by John Cremona that contains all isogeny classes of elliptic curves having conductor up to 130,000 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.

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 31 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 243. This database has been constructed by M. Grassl and the Magma group. The codes of length up to 21 are optimal.

Best Known F₄ Linear Codes: A database containing constructions of the best known linear codes over F₄ of length up to 256. This database has been constructed by M. Grassl and the Magma group. The codes of length up to 18 are optimal.

Best Known F₅, F₇, F₈, and F₉ Linear Codes: Similar databases exist for these codes. Their details and more information about the previous ones are summarised in the table below.

Irreducible Polynomials: A database of sparse irreducible polynomials, constructed by Allan Steel in 2004–2007. The polynomials have the form f(x) = xⁿ + g(x), where the degree of g is minimal and g is the first such polynomial in lexicographical order.

The database has the following degrees for the specified fields:

• GF(2): up to degree 120,000.

• GF(3): up to degree 50,000.

• GF(4), GF(5), GF(7): up to degree 20,000.

• GF(q), 9 ≤ q ≤ 127: up to degree 1,000 or more.

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. The database now extends up to degree 409 for p = 2 (with some gaps) and up to degree at least 4 for all p ≤ 109987.

Small Graphs: Several databases of small graphs are available, using data made available by Brendan McKay. These include:-

• All simple graphs on 2 to 10 vertices

• All connected simple graphs on 2 to 10 vertices

• All Eulerian graphs on 2 to 12 vertices

• All connected Eulerian graphs on 3 to 11 vertices

• Planar graphs on up to 11 vertices,

• Some self-complementary graphs of orders up to 20.

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.

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: This database contains information about almost simple groups G, where S ≤ G ≤ Aut(S) and S is a simple group. The groups G that are included in the database are those associated with S such that |S| is less than 16000000, as well as M₂₄, HS, J₃, McL, Sz(32) and L₆(2). In particular, the database includes all simple groups having a permutation representation of degree less than 1000. The groups in the database are defined on standard generators which can be used to create an isomorphism between an almost simple group in some arbitrary representation and the "standard" version of it stored in the database. The database was originally conceived by Derek Holt with a major extension by Volker Gebhardt and sporadic additions by Bill Unger. The following groups are included:

• Alternating groups: A_n for n ≤ 999;

• Classical groups: L₂(q), L₃(q), L₄(q) and L₅(q) for all prime powers q; L₆(3), L₇(3); L_d(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), O^m₈(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 Integral Matrices: This database contains representatives for all GL_n(ℤ)-conjugacy classes of irreducible maximal finite subgroups of GL_n(ℤ) for n < = 11 and n∈{13,17,19,23}.

Finite Groups of Rational Matrices: The maximal finite subgroups of GL(n,ℚ), for n up to 31.

Symplectic Matrix Groups: The maximal finite irreducible subgroups of Sp_2n(ℚ) for 1 ≤ i ≤ 11. These groups were classified by Markus Kirschmer who also constructed the database.

Quaternionic Matrix Groups: The finite absolutely irreducible subgroups of GL_n(D) where D is a definite quaternion algebra whose centre has degree d over ℚ and nd ≤ 10

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. Several matrices of dimensions between 36 and 60 were provided by D. Djokovic, including 1771 of degree 60.

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.

Database of Interesting Lattices: A database containing lattices of Sloane and Nebe, containing the automorphism group and Θ-series for many examples.

Solvable Lie algebras: All soluble Lie algebras of dimensions 2, 3 and 4 over any field are included in this database which was constructed by Willem de Graaf.

Nilpotent Lie algebras: This comprises all nilpotent Lie algebras having dimensions 3, 4, 5, and 6 over all base fields, except base fields of characteristic 2, when the dimension is 6. This database is due to Willem de Graaf.

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.

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.

Number Fields: Databases of some 2.6 million small number fields of degrees two to nine. These databases are searchable on discriminant range, signature, class number, class group, and Galois group.

Function Fields: Databases of various function fields with specified base finite field and degree. Supported combinations are: F₂, degrees 2 and 3; F₃, degree 2, F₄, degrees 2, 3, and 5; F₅, degrees 2, 3, 4, and 8; F₇, degree 9; F₁₁, degree 3; and F₁₃, degree 3.

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 S_n whose kernel has positive betti number.