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db-localfields.md

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Database localfields
Description Local number fields
Status production
Contact John Jones
Code local_fields
Collections fields

Todo:

  • Extend range?

Collection fields

  • _id: Assigned by mongo

Example: ObjectId('4e6647840eb55b3887000003')

  • n (integer): degree

Example: 6

  • p (integer): prime p for the base \Q_p

Example: 11

  • e (integer): ramification degree

Example: 3

  • f (integer): residue field degree

Example: 2

  • c (integer): valuation of the discriminant

Example: 4

  • label (string): label

Example: '11.6.4.1'

  • coeffs (list of integers): coefficients of a defining polynomial, starting with the constant term

Example: [41503, 0, 0, 220, 0, 0, 1]

  • aut (integer): number of automorphisms of the field

Example: 6

  • gal (pair of integers): the Galois group [n, t] for nTt

Example: [6, 2]

  • galT (integer): the T-number for the Galois group

Example: 2

  • inertia (four-tuple, degree, whether it is transitive or not, the pair [d,t] for the degree and t-number if transitive, and optionally an html display string): inertia subgroup

Example: [3, 'i', [3, 1], 'C3']

  • slopes (list of rational numbers as string): wild ramification slopes

Example: '[]'

  • t (integer): tame degree for the Galois closure

Example: 3

  • u (integer): degree of maximal unramified subfield of the Galois closure

Example: 2

  • gms (rational number as string): Galois mean slope

Example: '2/3'

  • hw (TeX string): Hasse-Witt invariant as a string

Example: '$1$'

  • rf (pair of integers): root field

Example: [1, 1]

  • unram (polynomial in t as a string): polynomial defining maximal unramified subfield

Example: 't^2 - t + 7'

  • eisen (polynomial as string, in variable y over Q_p(t)): Eisenstein polynomial defining relative extension of this field over the maxmial unramified subfield

Example: 'y^3 - 11*t^3'

Index information for collection fields

  • {'_id': 1}: Created by mongo
  • {'e': 1}: search by ramification index
  • {'f': 1}: search by residue field degree
  • {'label': 1}: search by label
  • {'metadata': 1}: Created by mongo
  • {'n': 1}: search by degree
  • {'p': 1}: search by prime
  • {'p': 1,'n': 1}: search by prime and degree