db-Lattices.md

Database Lattices

Description	Integral lattices
Status	production
Contact	Samuele Anni
Code	lattice
Collections	lat

Contibutors: Samuele Anni, Anna Haensch, Gabriele Nebe, and Neil Sloane

The data is coming from here.

Collection lat

• aut (int): size of automorphism group
• base_label (string): part of the label which is completely deterministic
• class_number (int): class number or genus of a lattice
• comments (string): comments and historical remarks
• density (string): density of a lattice
• det (int): determinant of a lattice
• dim (int): dimension of a lattice
• genus_reps (list of matrices with int entries): list of genus representatives (matrices)
• gram (matrix with int entries): Gram matrix of a lattice
• hermite(string): Hermite number of a lattice
• index (int): index of a lattice
• kissing(int): Kissing number of a lattice
• label(string): LMFDB label of a lattice
• level(int): level of a lattice
• minimum(int): lenght of the shortest vector
• name(list of strings): list of known names of a given lattice
• shortest (list of vectors of int): list of shortest vectors*
• theta_series (list of int): coefficients of the q-expansion of the theta series associated to a lattice*

*for the Leech lattice it is a list of strings

LMFDB label of an integral lattice in lat

The LMFDB label has the following structure:

dimension.determinant.level.class_number.number

the first 4 parameters are uniquely defined, given a lattice, the last is depending on the ordering we compute/upload the data

Indexes

• {'id': 1} (created by mongo db)
• {'aut_1': 1} (for searching by aut dimension)
• {'class_number_1': 1} (for searching by label)
• {'class_number_1_dim_1': 1} (for searching by class_number/dimension)
• {'det_1': 1}, (for searching by determiant)
• {'dim_1': 1}, (for searching by dimension)
• {'dim_1_det_1_level_1_class_number_1': 1} (for searching by dim/det/class_number)
• {'dim_1_det_1_level_1_class_number_1_label_1': 1} (for sorting search results)
• {'dim_1_label_1': 1} (for searching by dim/label)
• {'label_1': 1} (for searching by label)
• {'level_1': 1} (for searching by level)
• lat.rand (auxilliary collection used for random objection access)

One example of lat

  {
    "_id": "562cf752ffe979688b0a1f15", 
    "aut": 12, 
    "base_label": "2.3.3.1", 
    "class_number": 1, 
    "comments": "This is the $A2$ root lattice, also called the hexagonal lattice or triangular lattice", 
    "density": "0.906899682117108925297039128820", 
    "det": 3, 
    "dim": 2, 
    "genus_reps": [
      [
        [
          2, 
          1
        ], 
        [
          1, 
          2
        ]
      ]
    ], 
    "gram": [
      [
        2, 
        -1
      ], 
      [
        -1, 
        2
      ]
    ], 
    "hermite": "1.15470053837925152901829756100", 
    "index": 1, 
    "kissing": 6, 
    "label": "2.3.3.1.1", 
    "level": 3, 
    "minimum": 2, 
    "name": ["A2"], 
    "shortest": [
      [
        1, 
        0
      ], 
      [
        1, 
        1
      ], 
      [
        0, 
        1
      ]
    ], 
    "theta_series": [
      1, 
      0, 
      6, 
      0, 
      0, 
      0, 
      6, 
      0, 
      6, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      6, 
      0, 
      0, 
      0, 
      0, 
      0, 
      6, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      6, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      6, 
      0, 
      0, 
      0, 
      6, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      6, 
      0, 
      12, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      6, 
      0, 
      18, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0,
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      12, 
      0, 
      6, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      0, 
      12, 
      0, 
      0, 
      0, 
      6
    ]
  }