Richard Borcherds' Complete List of Even 25-Dimensional Lattices of Determinant 2

Part of the Catalogue of Lattices, which is a joint project of Gabriele Nebe, RWTH Aachen university ( and Neil J. A. Sloane, AT&T Labs-Research (

The Complete List of Even 25-Dimensional Lattices of Determinant 2

Richard Borcherds
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Cambridge, England
Email address

Taken from R. E. Borcherds, The Leech Lattice and Other Lattices, Ph. D. Dissertation, University of Cambridge, 1984.

Table -2.  The norm -2 vectors of II25,1.

The following sets are in natural 1:1 correspondence:

Orbits of norm -2 vectors in II25,1 under Aut(II25,1).
Orbits of norm -2 vectors u of D under Aut(D).
25 dimensional even bimodular lattices L.
L is isomorphic to u^. Table -2 lists the 121 elements of any of these three sets.

The height t is the height of the norm -2 vector u of D, in other words -(u,w) where w is the Weyl vector of D. The letter after the height is just a name to distinguish vectors of the same height, and is the letter referred to in the column headed ``Norm -2's'' of table -4. An asterisk after the letter means that the vector u is of type 1, in other words the lattice L is the sum of a Niemeier lattice and a1.

The column ``Roots'' gives the Dynkin diagram of the norm 2 vectors of L arranged into orbits under Aut(L). ``Group'' is the order of the subgroup of Aut(D) fixing u. The group Aut(L) is a split extension R.G where R is the Weyl group of the Dynkin diagram and G is isomorphic to the subgroup of Aut(D) fixing u.

``S'' is the maximal number of pairwise orthogonal roots of L.

The column headed ``Norm 0 vectors'' desribes the norm 0 vectors z corresponding to each orbit of roots of u^ where u is in D, as in 3.5.2. A capital letter indicates that the corresponding norm 0 vector is twice a primitive vector, otherwise the norm 0 vector is primitive. x stands for a norm 0 vector of type the Leech lattice. Otherwise the letter a, d, or e is the first letter of the Dynkin diagram of the norm 0 vector, and its height is given by height(u)-h+1 where h is the Coxeter number of the component of the Dynkin diagram of u.

For example, the norm -2 vector of type 23a has 3 components in its root system, of Coxeter numbers 12, 12, and 6, and the letters are e, a, and d, so the corresponding norm 0 vectors have Coxeter numbers 12, 12, and 18 and hence are norm 0 vectors with Dynkin diagrams E64, A11D7E6, and D10E72.

See 4.3 for more information.

Height Roots  Group   S Norm 0 vectors
1a* a1   8315553613086720000    1X
2a  a2   991533312000    1x
3a  a19   92897280    9a
4a  a2a112   190080    13aa
5a* a124a1   244823040    25aA
5b  a24a19   3456    13aa
5c  a3a115   40320    17aa
6a  a29   3024    9a
6b  a3a25a16   240    13aaa
7a* a212a1   190080    13aA
7b  a33a24a13   48    13aaa
7c  a34a18a1   384    17aad
7d  a4a26a15   240    13aaa
7e  d4a121   120960    25ad
8a  a36a2   240    13ad
8b  a4a33a23a12   12    13aaaa
8c  d4a29   864    13aa
9a* a38a1   2688    17aA
9b  a42a34a1   16    13aaa
9c  a43a3a22a13   12    13aaaa
9d  d4a34a3a13   48    17aada
9e  a5a33a24   24    13aaa
9f  a5a34a16   48    17aaa
10a  d4a43a23   12    13aaa
10b  a5a42a32a2a1   4    13aaaaa
11a* a46a1   240    13aA
11b  d44a19   432    25dd
11c  a5d42a33   24    17daa
11d  a5a5a42a3a1   4    13aaaaa
11e  a52d4a32a12a1   8    17aaaad
11f  a53a24   48    13aa
11g  d5a36a1   48    17aad
11h  a6a42a32a2a1   4    13aaaaa
12a  a54a2   24    13ad
12b  d5a44a2   8    13aaa
12c  a6d4a43   6    13aaa
12d  a6a52a3a22   4    13aaaa
13a* a54d4a1   48    17aaA
13b  d5a52d4a3a1   4    17aaada
13c  d5a53a13a1   12    17aaae
13d* d46a1   2160    25dD
13e  a62a5a4a12   4    13aaaa
13f  a7a5a42a3   4    13aaaa
13g  a7a5d4a32a12   4    17aaaaa
14a  a6a6d5a4a2   2    13aaaaa
14b  a63d4   12    13aa
14c  a7a6a5a4a1   2    13aaaaa
15a* a64a1   24    13aA
15b  d53a5a3   12    17ade
15c  d6d44a13   24    25ddd
15d  d6a52a5a3   4    17aada
15e  a7d52a32a1   4    17aaad
15f  a72d42a1   8    17aad
15g  a8a53   6    13aa
15h  a8a6a5a3a2   2    13aaaaa
16a  a73a2   12    13ad
16b  a8a6d5a4   2    13aaaa
17a* a72d52a1   8    17aaA
17b  e6a53d4   12    17aae
17c  a7d6d5a5   2    17daaa
17d  a72d6a3a1   4    17aada
17e  a8a72a1   4    13aaa
17f  a9d5a5d4a1   2    17aaaaa
17g  a9a7a42   4    13aaa
18a  e6a63   6    13aa
18b  a9a8a5a2   2    13aaaa
19a* a83a1   12    13aA
19b  d63d4a13   6    25ddd
19c  a7e6d52a1   4    17eaad
19d  d7a7d5a5   2    17aaad
19e  d7a72a3a1   4    17aaad
19f  a9a7d6a1a1   2    17aaaad
19g  a10a7a6a1   2    13aaaa
20a  a82e6a2   4    13aaa
20b  a10a8d5   2    13aaa
21a* a92d6a1   4    17aaA
21b  a11d6a5a3   2    17aaaa
21c  a11a8a5   2    13aaa
21d* d64a1   24    25dD
21e  a9e6d6a3   2    17aaad
23a  d7e62a5   4    17ead
23b  d8d62d4a1   2    25dddd
23c  a9d72   4    17da
23d  a9d8a7   2    17daa
23e  a11d7d5a1   2    17aaad
24a  a112a2   4    13ad
24b  a12e6a6   2    13aaa
25a* a11d7e6a1   2    17aaaA
25b  a13d6d5   2    17aaa
25e* e64a1   48    17eE
26a  a13a10a1   2    13aaa
27a* a122a1   4    13aA
27b  e7d63   3    25dd
27c  a9a9e7   2    17ada
27d  d9a9e6   2    17ada
27e  a11d9a5   2    17aad
27f  a14a9a2   2    13aaa
29a  a11e7e6   2    17daa
29d* d83a1   6    25dD
31a  d82e7a1a1   2    25ddde
31b  d10d8d6a1   1    25dddd
31c  a15d8a1   2    17aad
33a* a15d9a1   2    17aaA
33b  a15e7a3   2    17aad
33c  a17a8   2    13aa
35a  e73d4   6    25de
35b  a13d11   2    17da
36a  a18e6   2    13aa
37a* a17e7a1   2    17aaA
37d* d10e72a1   2    25ddD
39a  d12e7d6   1    25ddd
45d* d122a1   2    25dD
47a  d10e8e7   1    25edd
47b  d14d10a1   1    25ddd
47c  a17e8   2    17da
48a  a23a2   2    13ad
51a* a24a1   2    13aA
61d* d16e8a1   1    25ddD
61e* e83a1   6    25eE
63a  d18e7   1    25dd
93d* d24a1   1    25dD