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

A Table of the Highest Kissing Numbers Presently Known

A Table of Perfect Lattices

Unimodular Lattices, Including A Table of the Best Such Lattices

Modular Lattices, Including A Table of the Best Such Lattices

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

Root Lattices and Dual (or Weight) Lattices

• 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.

Laminated Lattices

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

Kleinian Lattices

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

• L20

1-Dimensional Lattices

• The integer lattice Z, or (equivalently) the root lattice A1

2-Dimensional Lattices

• The root lattices A2, A2*, D2, D2*. The last two are equivalent to the simple square lattice Z2

3-Dimensional Lattices

• 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.

4-Dimensional Lattices

• 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.

5-Dimensional Lattices

• The lattices (A5)+2, P5.2 = (A5)+3
• The root lattices D5, D5*, A5, A5*
• Gordon Nipp's table of 48,000 reduced regular primitive positive-definite quinary quadratic forms of discriminants up through 513.

6-Dimensional Lattices

• 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

7-Dimensional Lattices

• See also under perfect lattices above.
• The root lattices A7, A7*, D7, D7*, E7, another version of E7, a third version of E7, E7*, another version of E7*
• The lattices KAPPA7, KAPPA7*

8-Dimensional Lattices

• 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

9-Dimensional Lattices

• The laminated lattice LAMBDA9
• The lattices KAPPA9, KAPPA9*, KAPPA9.2
• The root lattices D9, D9*, A9, A9*

10-Dimensional Lattices

• 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*

11-Dimensional Lattices

• The laminated lattices LAMBDA11_MAX, LAMBDA11_MIN
• The lattices KAPPA11, KAPPA11*
• A lattice without a basis of minimal vectors
• The root lattices D11, D11*, A11, A11*
• dim11kis308, dim11kis422,

12-Dimensional Lattices

• 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,

13-Dimensional Lattices

• 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,

14-Dimensional Lattices

• 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,

15-Dimensional Lattices

• 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,

16-Dimensional Lattices

• 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

17-Dimensional Lattices

• 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*

18-Dimensional Lattices

• 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*

19-Dimensional Lattices

• Laminated lattice LAMBDA19
• The lattice KAPPA19
• Root lattices A19, A19*, D19, D19*

20-Dimensional Lattices

• 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*

21-Dimensional Lattices

• 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*

22-Dimensional Lattices

• 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

23-Dimensional Lattices

• The shorter Leech lattice O23
• The lattice Q_23(6)^{+2}
• Laminated lattice LAMBDA23
• Root lattices A23, A23*, D23, D23*

24-Dimensional Lattices

• 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*

25-Dimensional Lattices

• Laminated lattice LAMBDA25

• R. E. Borcherds's list of 25-Dim. unimodular lattices; also two explicit examples, with kissing numbers 2 and 92.

• R. E. Borcherds's list of even 25 -dimensional lattices of determinant 2.
• Lattices from the maximal finite subgroups of GL(25,Q) [see G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397]: A5 tilde(otimes) A5, C2xPSL(2,49):2

• The root lattice D25

26-Dimensional Lattices

• 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

27-Dimensional Lattices

• 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

28-Dimensional Lattices

• 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

29-Dimensional Lattices

• Laminated lattice LAMBDA29
• An extremal unimodular lattice in dimension 29

30-Dimensional Lattices

• 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

31-Dimensional Lattices

• Lattices from the maximal finite subgroups of GL(31,Q) [see G. Nebe, Finite subgroups of GL(n,Q) for 25 <= n <= 31. Comm. Algebra 24 (7) (1996), 2341-2397]: L2(32):5, L3(5):2

• Laminated lattice LAMBDA31
• A 31-dim. section of the Quebbemann lattice Q32
• An extremal unimodular lattice in dimension 31

32-Dimensional Lattices

• Barnes-Wall BW32, BW32 as a Hurwitzian lattice
• Quebbemann lattices Q32, Q32'
• Bachoc's 8-dimensional Hurwitzian lattice,
• Bachoc's 16-dimensional Eisenstein lattice,
• Cyclo-quaternionic lattices C((5+sqrt(17))/2)L_32,2, L_32,2, L_32,6

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.)

The 15 Koch-Venkov extremal 32-dimensional unimodular lattices:

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

• Quebbemann Q32
• Quebbemann Q32'
• Bachoc B32
• Cyclo-quaternionic CQ32
• A new lattice U32

Hurwitzian lattices: The 8 indecomposable P-modular (and real-unimodular) lattices

• LL32
• LL32'
• LL32''
• Barnes-Wall BW32
• L8(P)
• L8(P^2)
• L8(P^4)
• L8(P^8)

Hurwitzian lattices: The 15 indecomposable hermitian unimodular lattices of rank 8 (and real determinant 2^16)

• MxE8
• D8(1+i)
• D8(1-w)
• D4(1+i)^2
• S1^2
• MxD8
• D3(1+i) + Mx_wU5
• D2(1+i) + M x D6
• D4(1-w)^2
• MxA8
• D2(1+i)^4
• MxD4^2
• D3(1-w) + MxA5
• MxA4^2
• D1(1+i)^8

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)

• BW32
• SL2(5)^4:S4.2a
• SL2(5)^4:S4.2b
• 4.L3(4).2^2a
• 4.L3(4).2^2b
• SL2(17)SS3.2a
• SL2(17)SS3.2b
• 2.Alt7 2.Alt7.2^2
• 2.Alt7 SL2(3)C3.2 .2
• Sp4(3)Sp4(3):2 C3.2
• SL2(5)2^1+6.O6-(2).2
• SL2(9).2 2^1+4.Alt_5.2
• SL2(5) Sp4(3)C3.2 .2
• SL2(17)SS3.2
• SL2(7) 2.Alt7.2
• SL2(9) D10 SL2(5).2
• SL2(7) SL2(3)C3.2 .2
• SL2(7) SL2(3)C3.2 .2
• SL2(3)C4.2 SL2(7).2
• SL2(7) SL2(9).2
• SL2(9) SL2(5).2
• SL2(5)SL2(5):2 C3D8.2 .2a
• SL2(5)SL2(5):2 C3D8.2 .2b
• C15:C4 F4.2
• 2^1+4Alt5 SL2(5) D10.2
• SL2(5)D10 SL2(3)C3.2 .2
• SL2(5) C5D24.2
• SL2(3) C5D24.2 .2
• SL2(5) C5Q24.2
• SL2(3) C5Q24.2 .2

33-Dimensional Lattices

• Unimodular lattice with min norm 3

34-Dimensional Lattices

• Unimodular lattice with min norm 3
• Bachoc's 17-dimensional Eisenstein lattice,

35-Dimensional Lattices

• Unimodular lattice with min norm 3

36-Dimensional Lattices

• Harada's odd unimodular lattice with long shadow,
• Bachoc's 9-dimensional Hurwitzian lattice,
• Kschischang-Pasupathy lattice KP36
• lattice B_36 based on Rao-Reddy code
• Cyclo-quaternionic lattice L_36,4
• Cyclo-quaternionic lattice L_36,4sup1
• Cyclo-quaternionic lattice L_36,4sup2
• Unimodular lattice Sp4(4)D8.4 with min norm 4
• Another unimodular lattice with min norm 4
• Delta1 Delta2 Delta3 Delta4 Delta5 The five 3-modular lattices obtained by tripling CoxeterTodd
• 3-modular lattice from Sp4(4)

  Lattices listed in G. Nebe: Finite quaternionic matrix groups, Representation Theory 2, 106-223 (1998):

  • SL2(19)SL2(3).2
  • related 2-modular lattices SL2(19).2 SL2(19)C3
  • U3(3)U3(3):2.2

37-Dimensional Lattices

• Harada's odd unimodular lattice with long shadow,

38-dimensional Lattices

• Extremal unimodular 38-dim. lattice

39-dimensional Lattices

• Extremal unimodular 39-dim. lattice

40-dimensional Lattices

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):

• U3(5):3C3.2
• 5^1+2:SL2(5).2 SL2(5).2
• SL2(11) 2^1+4.Alt5 .2
• U5(2) 2^1+4.Alt5 .2 , Extremal even unimodular 40-dim. lattice
• 2.U4(2)SL2(3).2
• SL2(11)C12.2.2
• SL2(11)SL2(3).2
• SL2(19)SS3.2
• 2.Alt7SS3.2
• 2.Alt7SS3
• 2.U4(3).4SS3.2
• F4S6.2
• L2(11) SL2(3) S3 .2
• 2.M12.2 GL2(3)

Further 40-dimensional lattices

• Bachoc's 10-dimensional Hurwitzian lattice,
• Extremal odd unimodular 40-dim. lattice,
• Extremal 3-modular 40-dim. lattice related to M22

Higher-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

Other Links Related to Lattices

• Richard Borcherds' table of positive definite lattices.
• Jacques Martinet's home page.
• Peter Scholl, Sphere Packing Database (for packings of finitely many spheres).

ABBREVIATIONS

See also our home pages: Gabriele Nebe and Neil Sloane.