A Catalogue of Lattices

 This data-base of lattices is a joint project of Gabriele Nebe, RWTH Aachen university (nebe(AT)math.rwth-aachen.de) and Neil Sloane. (njasloane(AT)gmail.com).

 Our aim is to give information about all the interesting lattices in "low" dimensions (and to provide them with a "home page"!). The data-base now contains about 160,000 lattices!

 Tables: densest packings, kissing numbers, modular lattices, unimodular lattices.

 Classifications: Single class genera of integral lattices, Bravais lattices, Brandt-Intrau ternary forms, Gordon Nipp's tables of quaternary and quinary forms, Niemeier lattices, Borcherds's lists of 25-dim lattices, perfect lattices, laminated lattices.

 Lattices in : 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, and higher, dimensions.

 Root lattices: root lattices or weight lattices, more precisely An lattices, An* lattices, Dn lattices, Dn* lattices, E6, E7, E8 lattices and their duals,

 Documentation and Skripts
abbreviations, change library file in html format to standard format, change standard format to GAP format, change standard format to MACSYMA format, change standard format to MAGMA format, change standard format to MAPLE format, change standard format to PARI format, links

 Keywords: tables, lattices, quadratic forms, lattice packings, lattice coverings, An lattices, An* lattices, anabasic lattice, Barnes-Wall lattices, binary quadratic forms, body-centered cubic lattice, Borcherds's lists of 25-dim lattices, Brandt-Intrau ternary forms, Bravais lattices, contact numbers, Coxeter-Todd lattice, crystallographic lattices, densest packings, Dn lattices, Dn* lattices, E6, E7, E8 lattices and their duals, Eisenstein lattices, Elkies-Shioda lattices, face-centered cubic lattice, Hurwitzian lattices, isodual lattices, William Jagy: ternary forms that are spinor regular but not regular, kissing numbers, Kleinian lattices, Kschischang-Pasupathy lattices, laminated lattices, Leech lattice, mean-centered cubic lattice, modular lattices, Mordell-Weil lattices, Newton numbers, Niemeier lattices, Gordon Nipp's tables of quaternary and quinary forms, perfect lattices, Quebbemann lattices, Rao-Reddy code, root lattices, SPLAG, ternary quadratic forms, unimodular lattices, Single class genera of integral lattices, weight lattices, lattices in 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, and higher, dimensions,

Remarks

 For the format and for various programs to convert to other formats, see ABBREVIATIONS.

 A gzipped file containing all the .std files can be downloaded here (about 1 meg).

 Warning! Not all the entries have been checked!

 Most lattices can be described in many different ways, e.g. the face-centered cubic lattice can be described using three coordinates, as D3, or using four coordinates, as A3. Our policy is that different definitions (or scales) for the same lattice should be in different files. Inside any particular file everything should be on the same scale and should be consistent. The determinant given should be the determinant of the Gram matrix given in the file, and so on.

 Contributions of new lattices or additional information about the given lattices will be welcomed.

 Usually a star (*) denotes a dual lattice -- but in the file names "*" is replaced by an "s"; and in the two tables below "*" indicates a nonlattice packing that is better than any lattice presently known.

 As a general reference for the subject covered in this catalogue see SPLAG

 Note that the theta series of many of these lattices can be found in NJAS's On-Line Encyclopedia of Integer Sequences. The sequence 1, 6, 12, 8, 6, 24, 24, ... for example is the theta series of the simple cubic lattice.

 The data-base has also benefitted from contributions or suggestions from the following friends:
Richard Borcherds (R.E.Borcherds(AT)pmms.cam.ac.uk), John Conway (conway(AT)math.princeton.edu), Will Jagy (jagy(AT)msri.org), Irving Kaplansky (kap(AT)msri.org), Gordon Nipp (gnipp(AT)calstatela.edu), Richard Parker (richard(AT)ukonline.co.uk), Eric Rains (rains(AT)research.att.com), Alexander Schiemann (aschi(AT)math.uni-sb.de), Bernd Souvignier (bernd(AT)maths.usyd.edu.au), Allan Steel (allan(AT)maths.su.oz.au).

 A Table of the Densest Packings Presently Known

(In a separate file)

 A Table of the Highest Kissing Numbers Presently Known

(In a separate file)

 A Table of Perfect Lattices

(In a separate file)

 Unimodular Lattices, Including A Table of the Best Such Lattices

(In a separate file)

 Modular Lattices, Including A Table of the Best Such Lattices

(In a separate file)

Named Lattices

Root Lattices and Dual (or Weight) Lattices

Laminated Lattices

Reference: SPLAG Chap. 6.

The KAPPA_n Lattices

Reference: SPLAG Chap. 6.

Kleinian Lattices

That is, lattices over Z[(1+sqrt(-7))/2].

1-Dimensional Lattices

2-Dimensional Lattices

3-Dimensional Lattices

4-Dimensional Lattices

5-Dimensional Lattices

6-Dimensional Lattices

7-Dimensional Lattices

8-Dimensional Lattices

9-Dimensional Lattices

10-Dimensional Lattices

11-Dimensional Lattices

12-Dimensional Lattices

13-Dimensional Lattices

14-Dimensional Lattices

15-Dimensional Lattices

16-Dimensional Lattices