This data-base of lattices is a joint project of

Neil J. A. Sloane
(njasloane@gmail.com)
and

Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)

Our aim is to give information about all the interesting
lattices in "low" dimensions.

Contributions are welcome.

For a list of the lattices in the data-base, see the
**Index file**.

The files in the data-base are in two different formats:

(i) the standard format (with a file name such as E8.std), which
is what we use internally; these files are in plain ASCII,
which is the most convenient for computations; and

(ii) the html format (with a file name such as E8.html),
which looks better on the World Wide Web; this is the format you
will see when you download files from the data-base.

Several shell scripts are available for converting from the html format to other formats:

**html format to standard format**- this is important, since we assume that when you download a lattice from the data-base, you will immediately use this program to convert it to the standard format.**standard format to GAP format****standard format to MACSYMA format****standard format to MAGMA format****standard format to MAPLE format****standard format to PARI format**

Each lattice is specified by some subset of the following properties

(further properties are being added):

- %NAME (required)
- %DIMENSION (required)
- %DET (determinant)
- %TYPE
- %BASIS (basis for lattice; may have more than n coordinates)
- %TRIANGULAR_BASIS (triangular & square matrix giving basis for lattice)
- %GRAM (floating point or integer Gram matrix)
- %GRAM(MAPLE) (Gram matrix in Maple format)
- %GRAM(PARI) (Gram matrix in PARI format) (etc)
- %EIGENVALUES
- %DIVISORS (elementary divisors)
- %MINIMAL_NORM
- %DENSITY (center density)
- %HERMITE_NUMBER
- %KISSING_NUMBER
- %MINVECS
- %THETA_SERIES
- %COVERING_NORM (square of covering radius)
- %GLUE_VECTORS (representatives for L*/L )
- %GENUS
- %GROUP_ORDER
- %GROUP_NAME
- %GROUP_GENERATORS
- %HERMITIAN_STRUCTURE
- %HERMITIAN_GROUP_ORDER
- %HERMITIAN_GROUP_GENERATORS
- %PROPERTIES

INTEGRAL = 1 (lattices are real unless stated otherwise) - %BACHER_POLYNOMIAL
- %REFERENCES
- %NOTES
- %URL (links to other sites for this lattice)
- %LAST_LINE (required)

For an explanation of (most of) these terms, see SPLAG:

"SPLAG" is an abbreviation for the book
**Sphere Packings, Lattices and Groups**,

by J. H. Conway and N. J. A. Sloane, Springer-Verlag, NY, 1988;
3rd edition in preparation, 1996;

Russian translation published by Mir, Moscow, 1990.

For a description of the Bacher polynomials, click
**here**

Convention: 0 = FALSE, 1 = TRUE

(But click
**here**
to see the html version, which looks nicer!)

%NAME (obligatory) E8 %DIM (obligatory) 8 %BASIS (optional) 8 8 (no of rows, no of columns) then 64 numbers giving entries in generator matrix %TRIANGULAR_BASIS (optional) 8 8 (no of rows, no of columns) then 64 numbers giving entries in triangular and square generator matrix %GRAM (optional) 8 8 then 64 numbers giving entries of Gram matrix or %GRAM (optional) 8 0 then 8*9/2 = 36 numbers giving entries of Gram matrix in triangular format %DET (optional) 1 %GLUE_VECTORS (optional) nogl then nogl numbers for the glue vectors (including the 0 glue word) %PROPERTIES (optional) INTEGRAL=1 (means this is an integral lattice) %MINIMAL_NORM (optional) 2 (minimal norm) %KISSING_NUMBER (optional) 240 (kissing number, i.e. number of minimal vectors) %DENSITY (optional) .0625 (center density delta, Ref. SPLAG p. 13) %HERM (optional) 2 (Hermite number = 4 * delta^(2/dim) ) %NOTES (optional) comment lines go here %REFERENCES SPLAG p. 120. %LAST_LINE (obligatory last line)