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Database transitivegroups
Description Galois groups
Status production
Contact John Jones
Code galois_groups
Collections groups, Gmodules

Todo:

  • Make the collection groups information prettier

Collection groups

  • label (string): label is of the form nTt where n is the degree and t is the "t-number"
  • n (int): the degree (n from Sn)
  • t (int): the t-number, a standard index for conjugacy classes of subgroups of Sn
  • auts (int): the number of automorphisms a degree n field with this as its Galois group
  • order (int): the size of the group
  • parity (int): 1 if the group is a subgroup of An, otherwise -1
  • cyc (int): 1 if the group is cyclic, otherwise 0
  • solv (int): 1 if the group is solvable, otherwise 0
  • subs (list of pairs [n,t]): if K is a degree n field with this Galois group, this gives the subfields up to isomorphism in terms of their Galois groups
  • repns (list of pairs [n,t]): if K is a degree n field with this Galois group, this gives other small degree fields with the same Galois closure, up to isomorphism, in terms of their Galois groups
  • arith_equiv (int): number of arithmetically equivalent fields for number fields with this Galois group
  • resolve (list of pairs [n,t]):
  • name (string): the name given by gap (also used by pari, magma, sage, etc)
  • pretty (string): latex of a nicer name for this group

Index information for collection groups:

  • {'_id': 1}: mongo internal
  • {'n': 1, 't': 1}: for Galois group search
  • {'t': 1}: for Galois group search

Collection Gmodules

Each Gmodule is uniqely determined by a Galois group (nTt) and then an abitrary index.

  • _id (): Object id assigned by mongo

**Example**: 53dbc9290eb55b5881ba484f

* **complete** (int): Do we have complete information, 1 for yes, 0 for no

**Example**: 1

* **dim** (int): dimension of lattice

**Example**: 1

* **gens** (pair [string of cycle type, matrix for action]): generators of group and matrix for their actions

**Example**: [['(1,2,3)', [[1]]], ['(1,2)', [[1]]]]

* **n** (int): for Galois group in form nTt

**Example**: 3

* **t** (int): for Galois group in form nTt

**Example**: 2

* **index** (int): index to identify the G-module

**Example**: 0

* **name** (string): name of this G-module

**Example**: 'Triv'

Index information for collection Gmodules:

  • {'_id': 1}: mongo internal