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,
modular lattices,
extremal strongly modular lattices,
unimodular lattices.
External Tables:
kissing numbers
maintained by Henry Cohn
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,
strongly perfect 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,
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,
Kleinian lattices,
Kschischang-Pasupathy lattices,
laminated lattices,
Leech lattice,
mean-centered cubic lattice,
modular lattices,
extremal strongly modular lattices,
Mordell-Weil lattices,
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).
(In a separate file)
(In a separate file)
(In a separate file)
(In a separate file)
(In a separate file)
(In a separate file)
Named Lattices
- The Coxeter-Todd lattice K12
- Leech lattice
- Leech lattice as a Hurwitzian lattice
- Barnes-Wall lattices BW4 = D4,
BW4' = D4*,
BW8 = E8,
BW8' = E8_code,
BW16 = LAMBDA16,
BW16 as a Hurwitzian lattice,
the odd 16-dim Barnes-Wall lattice,
BW32,
BW32 as a Hurwitzian lattice
- Quebbemann lattices Q32,
Q32'
- Mordell-Weil lattice MW44,
Mordell-Weil lattice MW64
- The A_n lattices:
A1,
A2,
A3,
A4,
A5,
A6,
A7,
A8,
A9,
A10,
A11,
A12,
A13,
A14,
A15,
A16,
A17,
A18,
A19,
A20,
A21,
A22,
A23,
A24
- The A_n* lattices:
A1*,
A2*,
A3*,
A4*,
A5*,
A6*,
A7*,
A8*,
A9*,
A10*,
A11*,
A12*,
A13*,
A14*,
A15*,
A16*,
A17*,
A18*,
A19*,
A20*,
A21*,
A22*,
A23*,
A24*
- The D_n lattices:
D1,
D2,
D3,
D4,
D5,
D6,
D7,
D8,
D9,
D10,
D11,
D12,
D13,
D14,
D15,
D16,
D17,
D18,
D19,
D20,
D21,
D22,
D23,
D24
- The D_n* lattices:
D2*,
D3*,
D4*,
D5*,
D6*,
D7*,
D8*,
D9*,
D10*,
D11*,
D12*,
D13*,
D14*,
D15*,
D16*,
D17*,
D18*,
D19*,
D20*,
D21*,
D22*,
D23*,
D24*
- The E_n lattices and their duals:
E6,
E6*,
E7,
E7a (a second version of E7),
E7*,
E8,
E8 as a Hurwitzian lattice.
For other versions of E6, E6*, E7, E7* and E8 see under
6,
7 and
8 dimensional lattices below.
Reference: SPLAG Chap. 6.
- LAMBDA1 = A1,
LAMBDA2 = A2,
LAMBDA3 = D3,
LAMBDA4 = D4,
- LAMBDA5 = D5,
LAMBDA6 = E6,
LAMBDA7 = E7,
LAMBDA8 = E8,
- LAMBDA9,
LAMBDA10,
LAMBDA11_MAX,
LAMBDA11_MIN,
- LAMBDA12_MAX,
LAMBDA12_MID,
LAMBDA12_MIN,
- LAMBDA13_MAX,
LAMBDA13_MID,
LAMBDA13_MIN,
- LAMBDA14,
LAMBDA15,
LAMBDA16 = BW16,
- LAMBDA17,
LAMBDA18,
LAMBDA19,
LAMBDA20,
- LAMBDA21,
LAMBDA22,
LAMBDA23,
LAMBDA24 (The Leech lattice)
- LAMBDA25,
LAMBDA26,
LAMBDA27,
LAMBDA28
- LAMBDA29,
LAMBDA30,
LAMBDA31
- The table also contains many integral laminated lattices, including
these from the paper
W.Plesken, M.Pohst, Constructing integral lattices with prescribed minimum II,
Math. Comp. Vol 60 (1993), pp. 817-825:
LAMBDA14.2,
LAMBDA14.3,
LAMBDA14.4,
LAMBDA15.2,
LAMBDA15.3,
LAMBDA15.4,
LAMBDA16.2,
LAMBDA16.3,
LAMBDA16.4,
LAMBDA17.2,
LAMBDA17.3,
LAMBDA17.4,
LAMBDA18.2,
LAMBDA18.3,
The KAPPA_n Lattices
Reference: SPLAG Chap. 6.
- KAPPA7,
KAPPA7*,
KAPPA8,
KAPPA8*,
KAPPA9,
KAPPA9*,
KAPPA10,
KAPPA10*,
KAPPA10a,
KAPPA10a*,
- KAPPA11,
KAPPA11*,
KAPPA12 = K12 (The Coxeter-Todd lattice),
KAPPA13,
KAPPA13*
- KAPPA14,
KAPPA15,
KAPPA16,
KAPPA17,
KAPPA18,
KAPPA19,
KAPPA20,
- The table also contains the following "integral Kappa's"
from the paper
W.Plesken, M.Pohst, Constructing integral lattices with prescribed minimum II,
Math. Comp. Vol 60 (1993), pp. 817-825:
KAPPA8.2,
KAPPA9.2,
KAPPA14.2,
KAPPA15.2,
KAPPA16.2,
KAPPA16.3,
KAPPA17.2
That is, lattices over Z[(1+sqrt(-7))/2].
- Face-centered cubic lattice as
D3 or
as A3
- Body-centered cubic lattice
as D3* or
as A3*
-
"Mean-centered cuboidal" lattice,
the densest isodual lattice in 3 dimensions
- The single class genera of integral lattices
G.L. Watson classified all genera of integral lattices that consist of a single
isometry class. These hand computations have been automatized and completed by D. Lorch
Single-Class Genera of Positive Integral Lattices
The ascii files
Single class genera
contain the list of Gram matrices of the lattices, where the lower left
entries of the symmetric Gram matrix are given.
A MAGMA program to read these file is available
here
- The 14 Bravais lattices
We give both the classical holotype (the smallest determinant
of any classically integral lattice of the type) and
the even holotype (the smallest determinant
of any even lattice of the type):
cubic P (the simple cubic lattice),
cubic P (even),
cubic I,
cubic I (even),
cubic F,
cubic F (even),
hexagonal P,
hexagonal P (even),
tetragonal P,
tetragonal P (even),
tetragonal I,
tetragonal I (even),
trigonal (or rhombohedral) R,
trigonal (or rhombohedral) R (even),
digonal (or orthorhombic) P,
digonal (or orthorhombic) P (even),
digonal (or orthorhombic) C,
digonal (or orthorhombic) C (even),
digonal (or orthorhombic) I,
digonal (or orthorhombic) I (even),
digonal (or orthorhombic) F,
digonal (or orthorhombic) F (even),
monoclinic P,
monoclinic P (even),
monoclinic C,
monoclinic C (even),
triclinic P,
triclinic P (even),
- The Brandt-Intrau tables of primitive positive-definite
odd ternary quadratic forms
and
even ternary quadratic forms
of discriminants up to -1000,
as recomputed by Alexander Schiemann.
- William C. Jagy,
Table of integer coefficient positive
ternary quadratic forms that are spinor regular but are not regular.
- The root lattices D4,
F4 (the same lattice as D4),
D4*,
A4,
A4*
- D4 as a Hurwitzian lattice
- The 4-D simple cubic lattice
- 4-d dual extremal lattices:
det 5.5,
det 11.11,
- 4-d modular lattices:
5-modular,
strongly 6-modular in "-" genus,
strongly 6-modular in "+" genus,
7-modular,
11-modular,
strongly 14-modular,
strongly 15-modular in genus "-",
strongly 15-modular even lattice E(15)
in genus "+",
strongly 15-modular odd associate O(15)
of previous lattice ,
21-modular lattice
23-modular lattice
- Gordon Nipp's table
of 74,000 reduced regular primitive positive-definite
quaternary quadratic forms of discriminants up through 1732.
- The root lattices A6 = P6.7,
A6*,
D6 = P6.3,
D6*,
E6 = P6.1,
another version of E6,
E6* = P6.2
- The other perfect lattices in six dimensions:
P6.4 = A6,2,
P6.5 = A6 sup (2),
P6.6 = A6,1
- Other 6-dimensional lattices:
M6,2 = Q_6(4)^{+2} = F_15,
the unique indecomposable lattice of det 8,
- Some modular lattices:
a 3-modular lattice of minimal norm 2,
a 7-modular lattice of minimal norm 3, O(7),
a strongly 8-modular lattice of minimal norm 4,
an 11-modular lattice of minimal norm 4 (in the "-" genus),
an 11-modular lattice of minimal norm 4 (in the "+" genus),
a strongly 14-modular lattice of minimal norm 4,
a 23-modular lattice of minimal norm 7
- The root lattices A8,
A8*,
D8,
D8*,
E8,
another version of the root lattice E8
- The coding theory version of E8
- this and the root lattice version of E8 are the two 8-dim. Barnes-Wall lattices.
- E8 as a Hurwitzian lattice
- The lattices KAPPA8,
KAPPA8*,
KAPPA8.2
- Dual extremal lattices
det 6.6,
det 10.10,
- Other integral lattices:
A2 tensor D4,
H4,
(A2 X A4)*,
M8,3,
odd 5-modular lattice O(5),
odd 6-modular lattice O(6),
extremal 11-modular lattice in genus "-"
extremal 11-modular lattice in genus "+"
extremal strongly 14-modular lattice in genus "+"
extremal strongly 15-modular lattice in genus "+"
extremal strongly 15-modular lattice in genus "+"
-
See
Jacques Martinet's home page
for a list of the known perfect lattices in 8 dimensions.
- 8-Dim. isodual lattice from Bring curve
- The laminated lattice LAMBDA10
-
The lattices KAPPA10,
KAPPA10*
-
The lattice KAPPA10',
KAPPA10'*
- The shorter Coxeter-Todd lattice
- The lattices (C6 x SU(4,2)):C2,
C2 x S6,
A10^(2),
A10^(3)
- A 2-modular lattice,
a 4-modular lattice (Q10),
a 5-modular lattice in the "-" genus.,
a 5-modular lattice in the "+" genus.
- The root lattices D10,
D10*,
A10,
A10*
- The Coxeter-Todd lattice K12
- The laminated lattices
LAMBDA12_MAX,
LAMBDA12_MID,
LAMBDA12_MIN
- Lattices from the maximal finite subgroups of GL(12,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
C6.PSU(4,3).(C2 x C2),
((3+^(1+2):SL(2,3)) x SL(2,3)).C2,
(C2 x D10 x A5):C2,
(SL(2,5) Y SL(2,3)).C2,
A2xM6,2,
(C2 x C3.Alt6).(C2 x C2),
(PSL(2,7) x D8):C2,
(PSL(2,7) x D8):C2,
A2xA6,
A2 x A6^(2)
- Dual extremal lattices
det 5.5a,
det 5.5b,
det 11.11a,
det 11.11b,
- The unimodular lattice D12+
- The complete list of even 3-modular lattices in 12 dimensions:
The Coxeter-Todd lattice K12 as a modular
lattice,
A1^6.sqrt(3)A1^6,
A2^3.sqrt(3)A2^3,
A3^2.sqrt(3)A3^2,
A6.sqrt(3)A6,
D6.sqrt(D6),
E6.sqrt(3)E6,
G2.A5.sqrt(3)A5,
G2^2.D4.sqrt(3)D4,
G2^6
-
An extremal 5-modular lattice.
-
An extremal strongly 6-modular lattice.
- The odd Coxeter-Todd lattice
- Lift ternary Golay code to Z9 and
apply Construction A;
a different lift of the same code
- The root lattices D12,
D12*,
A12,
A12*
-
an odd extremal 5-modular lattice constructed by Xiaolu Hou and
Frederique Oggier
dim12modular5Group57600,
- An even 11-modular lattice of minimum 6
-
dim12kis462min8det5619712,
- The laminated lattices
LAMBDA13_MAX,
LAMBDA13_MID,
LAMBDA13_MIN
- The lattices KAPPA13,
KAPPA13*
- The lattices C2 x L(3,3):C2 = Q'_13(4)^{+2},
C2 x PSL(2,25):C2 = Q_13(2)^{+2}
- The root lattices D13,
D13*,
A13,
A13*
-
dim13kis526min8det14601600,
dim13kis672min8det11239424,
dim13kis720min8det10616832,
- The laminated lattice LAMBDA14
- The unimodular lattice E7^2+
- The extremal 14-dimensional 7-modular lattice
- A 14-dimensional even 11-modular lattice of minimum 6
- A 14-dimensional even 11-modular lattice of minimum 6
- A 14-dimensional odd 11-modular lattice of minimum 7
- A 14-dimensional odd 11-modular lattice of minimum 7
- Lattices from the maximal finite subgroups of GL(14,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
LAMBDA14.2,
LAMBDA14.3,
LAMBDA14.4,
KAPPA14,
KAPPA14.2
- The lattices C2 x G(2,3),
(SU(3,3) x C4).C2,
A2 x E7,
C2 x S7,
C2 x S8,
M14,2,
M14,3,
M14,6
- The root lattices D14,
D14*,
A14,
A14*
-
dim14kis486min36,
dim14kis522min28,
dim14kis552min16det488175316992,
dim14kis552min24,
dim14kis630min14det75614212800,
dim14kis654min12det8197085952,
dim14kis774min8det25021632,
- The laminated lattice LAMBDA15
- The lattices
LAMBDA15.2,
LAMBDA15.3,
LAMBDA15.4,
KAPPA15,
KAPPA15.2
- Lattices from the maximal finite subgroups of GL(15,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
Lambda2(E6),
C2 x Sp(6,2),
Lambda2(A6)
- The unimodular lattice A15+
- The root lattices D15,
D15*,
A15,
A15*
-
dim15kis1890min4det768,