Catalogue of Lattices

This data-base of lattices is a joint project of
Neil J. A. Sloane ( and
Gabriele Nebe, RWTH Aachen University (

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:

Each lattice is specified by some subset of the following properties
(further properties are being added):

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

Example: The E8 lattice

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

%NAME	 	(obligatory)
%DIM   		(obligatory)
%BASIS 	 	(optional)
8 8 		(no of rows, no of columns)
then 64 numbers giving entries in generator matrix
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


%GRAM		(optional)
8 0
then 8*9/2 = 36 numbers giving entries of Gram matrix in triangular format

%DET  		(optional)
%GLUE_VECTORS  		(optional)
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
SPLAG p. 120.
%LAST_LINE 		(obligatory last line)

Index file