Mathematical databases

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:

• Cunningham Factorizations: A database containing 221,188 factors f of numbers $a^n \pm 1$, where a < 10000, n < 10000, and f > 10⁹. The factorizations of integers of the form $a^n \pm 1$, $a \le 12$, were produced by Sam Wagstaff and collaborators. The factors for $13 \le a \le 99$ are mainly from the Brent-Montgomery-te Riele extension of the Cunningham tables. For 100 < a < 1000 they are mainly from the tables produced by Hisanori Mishima and Mitsuo Morimoto with additions from Pete Moore and Andy Steward.
• Elliptic Curves: The database of all elliptic curves of conductor up to 10,000, constructed by Cremona.
• 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.
• 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.
• Perfect Groups : The database of perfect groups of order up to a million constructed by Holt and Plesken.
• Almost Simple Groups: A database of almost simple groups of order less than $1.6 \times 10^8$ stored with their automorphism groups and maximal subgroups.
• 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.
• Transitive Groups: The transitive permutation groups of degree up to 22. The transitive groups of degree up to 15 were determined by Butler and McKay while the classification was extended to degree 22 recently by Hulpke.
• Primitive Groups: The table of primitive groups of degree up to 50 prepared by Sims.
• Irreducible Soluble Groups: The irreducible soluble subgroups of GL(n, p) for n > 1 and pⁿ < 256, as classified by Short.
• 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.
• Finite Groups of Rational Matrices: The maximal finite subgroups of $GL(n, \mbox{\bf Q})$ for n up to 31.
• Quaternionic Matrix Groups: The finite absolutely irreducible subgroups of $\mbox{\rm GL}_n({\cal D})$ where ${\cal D}$ is a definite quaternion algebra whose centre has degree d over $\mbox{\bf Q}$ and $nd\leq10$
• 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. R. Bülow.
• Irreducible polynomials: A database of sparse irreducible polynomials over $\mbox{\rm GF}(2)$ for all degrees up to 11000.
• Best Known 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 Mgama 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 best-known binary codes.