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:
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,
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)
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*,
- 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 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 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*
-
dim12kis462,
- 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*
-
dim13kis526,
dim13kis672,
dim13kis720,
- The laminated lattice LAMBDA14
- The unimodular lattice E7^2+
- The extremal 14-dimensional 7-modular lattice
- 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*
-
dim14kis486,
dim14kis522,
dim14kis552,
dim14kis588,
dim14kis630,
dim14kis654,
dim14kis774,
- 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*
-
dim15kis1872,
- Laminated lattice LAMBDA16 = Barnes-Wall BW16,
another version of BW16;
BW16 as a Hurwitzian lattice;
the odd Barnes-Wall lattice;
Overlattice of the Barnes-Wall lattice of minimum 3;
- The lattices
LAMBDA16.2,
LAMBDA16.3,
LAMBDA16.4,
KAPPA16,
KAPPA16.2,
KAPPA16.3
- Lattices from the maximal finite subgroups of GL(16,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
The lattices (SL(2,9) Y SL(2,9)).(C2 x C2),
E8 x A2,
((Sp(4,3) x C3) Y SL(2,3)).C2,
(((SL(2,5) Y SL(2,5)):C2) x D10):C2,
C2 x (S5 x S5):C2,
C2.A10,
A4 x F4,
(SL(2,5) Y (D8 Y Q8).A5).C2,
A2xH4,
(SL(2,5) Y SL(2,9)):C2,
(C2 x Alt6).(C2 x C2),
(SL(2,5) Y ((SL(2,3) x C3).C2)).C2,
D120.(C4 x C2),
(SL(2,7) Y C2.S3).C2,
C2 x S3 x PGL(2,7),
(C2.Alt7 Y C2.S3).C2,
(SL(2,7) Y C2.S3).C2,
D120.C2,
D120.C2.b,
A16^(3)
- Root lattices A16,
A16*,
D16,
D16*
- Dual extremal lattices
det 6.6a,
det 6.6b,
det 10.10a,
det 10.10b,
det 10.10c,
det 10.10d,
det 10.10e,
det 10.10f,
det 10.10g,
- An extremal 16-dimensional 7-modular lattice
- Lattices from the maximal finite subgroups of GL(14,Q) [see
- Laminated lattice LAMBDA17
- The lattices
LAMBDA17.2,
LAMBDA17.3,
LAMBDA17.4,
KAPPA17,
KAPPA17.2
-
The eight lattices associated with the group C2 x L(2,16):C4,
namely
Q_17(6),
Q_17(6)^{+2},
Q_17(6)^{+3},
Q_17(6)^{+6},
Q'_17(6),
Q'_17(6)^{+2},
Q'_17(6)^{+3},
Q'_17(6)^{+6}
- Root lattices A17,
A17*,
D17,
D17*
- Laminated lattice LAMBDA18
- The lattices
LAMBDA18.2,
LAMBDA18.3,
KAPPA18
- Lattices from the maximal finite subgroups of GL(18,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
(C2 x Sp(4,4)).C2,
(C2 x 3^(1+4):Sp(4,3)).C2,
(C2 x Alt5 x Alt5).(C2 x C2),
(C2 x C3.Alt6).(C2 x C2),
A2xA9,
(C2 x PSL(2,7) x PSL(2,7)).(C2 x C2),
M18,2,
M18,4,
A18^(5)
- Root lattices A18,
A18*,
D18,
D18*
- Lattices from the maximal finite subgroups of GL(20,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
(SU(5,2) x SL(2,3)).C2,
C2.M12.C2,
(D8 x S6).C2,
F4 x A5,
(C2 x SU(4,2)).C2,
(SU(4,2) x C6).C2,
C2 x 5^(1+2):GL(25),
A4 x A5,
(C2.PSL(3,4)).(C2 x C2),
C2.M22.C2,
C2 x S7,
(C2 x PSL(3,4)).(C2 x S3),
C2 x S8,
(PSL(2,11) x D12).C2,
(PSL(2,11) x D12).C2,
(SL(2,11) Y SL(2,3)).C2,
A2 x A10,
C2 x S3 x PGL(2,11),
C2 x S3 x PGL(2,11),
M20,3
- Dual extremal lattices
det 2.2a,
det 2.2b,
det 3.3,
det 5.5a,
det 5.5b,
det 11.11,
- Kleinian lattice L20 with group 2.M22.2
- Hurwitzian lattice L5(P^5)
- Hurwitzian lattice R20
- Cyclo-quaternionic lattice L_20,4
- Cyclo-quaternionic lattice
A_11.otimes(3).A_2
- Laminated lattice LAMBDA20
- The lattice KAPPA20
- The three extremal 2-modular lattices:
(SU(5,2) x SL(2,3)).C2,
C2.M12.C2,
HS20
- Root lattices A20,
A20*,
D20,
D20*
- Lattices from the maximal finite subgroups of GL(21,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
Lambda2(E7),
(C2 x PSU(4,3)).D8,
C2 x Sp(6,2),
(C2 x PSU(3.5)).S3,
C2 x S7
- Laminated lattice LAMBDA21
- Root lattices A21,
A21*,
D21,
D21*
- Lattices from the maximal finite subgroups of GL(22,Q) [see
G. Nebe, W. Plesken, Finite rational matrix groups. AMS-Memoir No. 556, vol. 116 (1995)]:
(C2 x PSU(6,2)).S3,
A2 x A11,
(C2 x HS).C2,
(C2 x Mc).C2,
hA22^(2),
hA22^(3),
A22^(4),
A22^(6)
- Laminated lattice LAMBDA22
- Root lattices A22,
A22*,
D22,
D22*
- Extremal 3-modular lattice of minimum 4
- Leech lattice LAMBDA24;
another version of Leech;
Leech lattice as a Hurwitzian lattice;
the odd Leech lattice
- The Niemeier Lattices (see SPLAG Table 16.1).
Here are the 23 Niemeier lattices, labeled by their root system:
D24,
D16_E8,
3E8,
A24,
2D12,
A17_E7,
D10_2E7,
A15_D9,
3D8,
2A12,
A11_D7_E6,
4E6,
2A9_D6,
4D6,
3A8,
2A7_2D5,
4A6,
4A5_D4,
6D4,
6A4,
8A3,
12A2,
24A1
- Hurwitzian lattices L6(P^6),
J24,
R24
- Cyclo-quaternionic (extremal 3-modular)
lattice C((5+sqrt(13))/2)L_24,2
- An optimal 11-modular
lattice (4+sqrt(5)) x Leech
- Lattices from the maximal finite subgroups of GL(24,Q) [see
G. Nebe, Finite subgroups of GL(24,Q). Exp. Math. Vol. 5, Number 3 (1996), 163-195]:
(((SL(2,5) Y SL(2,5)):C2) x Alt5).C2,
(C6.PSU(4,3).C2 Y SL(2,3)).C2,
((C2 x C3.Alt6).C2 Y SL(2,3)).C2,
(Sp(4,3) x 3^(1+2):SL(2,3)).C2,
F4xE6,
(C3.S6 x D8).C2,
(C6.PSL(3,4).C2 Y D8).C2,
((SL(2,3) Y C4).C2 x PSU(3,3)).C2,
(C2.J2 Y SL(2,5)):C2,
(SL(2,5) Y (D8 Y Q8).A5).C2,
(((SL(2,5) Y SL(2,5)):C2) x A5):C2,
W(F4) x S5,
(SL(2,5) Y (C2 x 3^(1+2)).GL(2,3)).C2,
S3 x (SL(2,5) Y SL(2,3)).C2,
(PSL(2,7) x W(F4)).C2,
(PSL(2,7) x W(F4)).C2,
F4 x A6,
W(F4) x PGL(2,7),
(SL(2,13) Y SL(2,3)).C2,
(SL(2,7) x PSL(2,7)).C2,
C6.Alt7:C2,
(C3.M10 x SL(2,3)).C2,
(Alt5 x ((C3 x D8).C2)).C2,
(C3.M10 x D8).C2,
S3 x ((PSL(2,7) x D8).C2),
S3 x ((PSL(2,7) x D8).C2),
((C2 x PSL(3,3)).C2 x C3).C2,
A2 x A12,
(C2 x D78).C12,
A4 x E6,
(C2 x C3.PGL(2,9) x D10).C2,
S3 x (C2 x D10 x A5).C2,
(C2 x PSU(4,2)).C2,
SL(2,7) Y (C2.S4),
(SL(2,7) Y Q16).C2,
A4xA6,
C2 x S5 x PGL(2,7),
(SL(2,11) Y SL(2,3)).C2,
C2 x PSL(2,11):C2
- Root lattices A24,
A24*,
D24,
D24*
- Bachoc's 13-dimensional Eisenstein lattice,
- Laminated lattice LAMBDA26
- The Conway-Borcherds unimodular lattices S26
and T26
- Lattices from the maximal finite subgroups of GL(26,Q)
[See
G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397.]
L4(3):2,
S6(3)C3.2,
S4(5):2,
L2(25):2^2,
L2(25):2,
L2(25):2
- Laminated lattice LAMBDA27
- Borcherds's unimodular lattice T27 with
minimal norm 3.
- Lattices from the maximal finite subgroups of GL(27,Q) [see
G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397]:
S9,
L3(3):2
- Bachoc's 14-dimensional Eisenstein lattice,
- Hurwitzian lattice L7(P^7)
- Hurwitzian lattice L7(P^3)
- Hurwitzian lattice L7(P)
- Hurwitzian lattice LL28
- Hurwitzian lattice R28
- Hurwitzian lattice R28'
- Hurwitzian lattice R28''
- Lattices from the maximal finite subgroups of GL(28,Q) [see
G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397]:
Sp6(3)C3.2,
2.J2YSL(2,3).2,
O8+(2):S3,
Sz(8):3YC4,
J2.2,
G2(3)xS3.2,
U3(3)(Q8C4).S3.2,
U3(5):2,
S8,
J2:2,
SL2(13)YSL2(3).2,
L2(13)S3.2
- Dual extremal lattice
det 3.3,
- Laminated lattice LAMBDA28
- Extremal 2-modular lattice of minimum 4
- Extremal 5-modular lattice of minimum 8
- Lattices from the maximal finite subgroups of GL(30,Q) [see
G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397]:
U4(2):2,
3.U4(3).2^2,
U4(2)3^1+2:SL2(3).2,
3.Alt6.2^2,
3.L3(4).2^2,
3.S7,
M30,2,
hat(A)30^(4),
hat(A)30^(8)
- Laminated lattice LAMBDA30
- A 30-dim. section of Quebbemann lattice Q32
32-dimensional even unimodular lattices
These have not yet been classified, and perhaps never will be.
However, the
mass
(the sum of reciprocals of orders of automorphism groups)
of all inequivalent 32 dimensional even unimodular lattices having any prescribed
root system has been determined by Oliver King (king(AT)math.berkeley.edu). (Root systems which aren't listed have mass zero.)
LAMBDA(RM)=BW32,
LAMBDA(QR),
LAMBDA(G),
LAMBDA(F),
LAMBDA(U),
LAMBDA(C1),
LAMBDA(C2),
LAMBDA(C3),
LAMBDA(C4),
LAMBDA(C5),
LAMBDA(G1),
LAMBDA(G2),
LAMBDA(G3),
LAMBDA(G4),
LAMBDA(S3)
Extremal 2-modular lattices
Hurwitzian lattices: The 8 indecomposable P-modular (and real-unimodular) lattices
Hurwitzian lattices: The 15 indecomposable hermitian unimodular lattices
of rank 8 (and real determinant 2^16)
Lattices of the maximal finite subgroups of GL(32,Q) containing
a maximal finite quaternionic matrix group as listed in
G. Nebe: Finite quaternionic matrix groups,
Representation Theory 2, 106-223 (1998)
Lattices of the maximal finite subgroups of GL(40,Q) containing
a maximal finite quaternionic matrix group as listed in
G. Nebe: Finite quaternionic matrix groups,
Representation Theory 2, 106-223 (1998):
Further 40-dimensional lattices
- An extremal unimodular lattice in dimension 42
- Another
- An extremal unimodular lattice in dimension 43
- Bachoc's 11-dimensional Hurwitzian lattice
- Mordell-Weil 44-dim. lattice MW44
- Cyclo-quaternionic 44-dim. lattice hat(A)_22^(2).otimes(3).A_2
- An extremal 44-dim. unimodular lattice
- An extremal 45-dim. unimodular lattice
- An extremal unimodular 46-dim. lattice
- An extremal unimodular 47-dim. lattice
- The extremal unimodular 48-dim.
lattice P_48p
- The extremal unimodular 48-dim.
lattice P_48q
- The extremal unimodular 48-dim.
cyclo-quaternionic lattice P_48n
-
a 5-modular lattice of minimum 12 obtained from P_48n
-
a 11-modular lattice of minimum 18 obtained from P_48n
-
An extremal 2-modular lattice obtained from P_48n
- An extremal 2-modular 48-dim. lattice
- Bachoc's 12-dimensional Hurwitzian lattice
- Turyns construction applied to extremal 5-modular lattice of dimension 16 yields a 5-modular lattice of dimension 48 with minimum 12
- A 3-modular 48-dim. lattice of minimum 8
- An extremal unimodular 52-dim. lattice
- An extremal unimodular 54-dim. lattice
- Cyclo-quaternionic 56-dim. lattices
L_56,2^(7)(M)
L_56,2(M)
L_56,2(tilde(M))
- An extremal unimodular 56-dim. lattice
- An extremal 60-dim. unimodular lattice
- Another extremal 60-dim. unimodular lattice (probably equivalent to the above)
- Cyclo-quaternionic 64-dim. lattices
L8,2.otimes.L_32,2
C(p3).L8,2.otimes.L_32,2
- Mordell-Weil 64-dim. lattice MW64
- An extremal 68-dim. unimodular lattice
- Another extremal 68-dim. unimodular lattice (probably equivalent to the above)
- An extremal unimodular 72-dim.
lattice
- Cyclo-quaternionic 72-dim. lattice
L24,2.otimes.L_24,2
-
Fi1,
Fi2:
Two 78-dim. lattices invariant under the Fischer group,
constructed by
Bernd Schroeder
- Extremal unimodular 80-dim. lattice L80
related to M22
- Extremal unimodular 80-dim. lattice M80 related to M22
- Shimada's 86-dimensional lattice
- Richard Parker's 105-dim. lattice obtained from A9
- Thompson-Smith 248-dimensional lattice
ABBREVIATIONS
See also our home pages:
Gabriele Nebe
and
Neil Sloane.