Classification of positive definite lattices. 22 Feb 2000

Richard E. Borcherds, ^{∗}

Mathematics department, Evans Hall #3840, University of California at Berkeley, CA 94720-3840 U.S.A.

e-mail: reb@math.berkeley.edu

www home page www.math.berkeley.edu/˜ reb

Contents.

1. An algorithm for classifying vectors in some Lorentzian lattices.

2. Vectors in the lattice II_{1,25}.

3. Lattices with no roots.

Table 0: Primitive norm 0 vectors in II_{1,25}.

Table 1: Norm 2 vectors in II_{1,25}. Table 2: Norm 4 vectors in II_{1,25}.

1. Classification of positive norm vectors.

In this paper we describe an algorithm for classifying orbits of vectors in Lorentzian lattices. The main point of this is that isomorphism classes of positive definite lattices in some genus often correspond to orbits of vectors in some Lorentzian lattice, so we can classify some positive definite lattices. Section 1 gives an overview of this algorithm, and in section 2 we describe this algorithm more precisely for the case of II_{1,25}, and as an application we give the classification of the 665 25-dimensional unimodular positive definite lattices and the 121 even 25 dimensional positive definite lattices of determinant 2 (see tables 1 and 2). In section 3 we use this algorithm to show that there is a unique 26 dimensional unimodular positive definite lattice with no roots. Most of the results of this paper are taken from the unpublished manuscript [B], which contains more details and examples. For general facts about lattices used in this paper see [C-S], especially chapters 15–18 and 23–28.

Some previous enumerations of unimodular lattices include Kneser’s list of the unimodular lattices of dimension at most 16 [K], Conway and Sloane’s extension of this to dimensions at most 23 [C-S chapter 16], and Niemeier’s enumeration [N] of the even 24 dimensional ones. All of these used some variation of Kneser’s neighborhood method [K], but this becomes very hard to use for odd lattices of dimension 24, and seems impractical for dimension at least 25 (at least for hand calculations; computers could probably push this further). The method used in this paper works well up to 25 dimensions, could be pushed to work for 26 dimensions, and does not seem to work at all beyond this.

We use the “(+, −, −, ⋯, −)” sign convention for Lorentzian lattices L, so that the reflection we are interested in are (usually) those of negative norm vectors of L. We fix one of the two cones of positive norm vectors and call it the positive cone. The norm 1 vectors in the positive cone form a copy of hyperbolic space in the usual way. We assume that we are given a group G of automorphisms of a Lorentzian lattice L, such that G is the semidirect product of a normal subgroup R generated by reflections of some negative

∗ Supported by NSF grant DMS-9970611. norm vectors, and a group Aut(D) of automorphisms preserving a fundamental domain domain D of R in hyperbolic space. We assume that all elements of u ∈ L having nonnegative inner product with all simple roots of R have norm (u, u) at least 0 (this is just to eliminate some degenerate cases). If L is a lattice then L(−1) is the lattice L with all norms multiplied by −1. We use Conway’s convention of using small letters a_{n}, d_{n}, e_{n} for the spherical Dynkin diagrams, and capital letters A_{n}, D_{n}, E_{n} for the corresponding affine Dynkin diagrams. The Weyl vector of a root system is the vector ρ such that (ρ, r) = −r^{2}/2 for any simple root r.

We want to find the orbits of positive norm vectors of the positive cone of L under the group G. Every positive norm vector of the positive cone of L is conjugate under R to a unique vector in D, so it is enough to classify orbits of vectors u in D under Aut(D).

The algorithm works by trying to reduce a vector u of D to a vector of smaller norm by adding a root of u^{⊥} to u. There are three possible cases we need to consider:

(1) There are no roots in u^{⊥}.

(2) There is a root r in u^{⊥} such that u + r ∈ D.

(3) There is at least one root in u^{⊥}, but if r is a root in u^{⊥} then u + r is never in D. We try to deal with these three cases as follows. If there are no roots in u^{⊥}, then we assume that D contains a non-zero vector w such

that (r, w) ≤ (r, u) for any simple root r and any vector u ∈ L in the interior of D. Then u − w has inner product at least 0 with all simple roots, so it also lies in D and has smaller norm than u unless u is a multiple of w and w^{2} = 0. So we can reduce u to a vector of smaller norm in D. The existence of a vector w with these properties is a very strong condition on the lattice L. Example 1.1. The lattices II_{1,9} and II_{1,17} have properties 1 and 2; this follows easily from Vinberg’s description [V85] of their automorphism groups. Conway showed that the lattice II_{1,25} also has these properties; see the next section. The lattices II_{1,8n+1} for n ≥ 4 do not have these properties; but the Minkowski-Siegel mass formula shows that these lattices have such vast numbers of orbits of positive norm vectors that there seems little point in classifying them. Example 1.2. It follows from [B90] that several lattices that are fixed points of finite groups acting on II_{1,25} also have a suitable vector w. For example the lattice II_{1,1} ⊕ BW (−1), where BW is the Barnes-Wall lattice, has this property. Some of the norm 0 vectors correspond to the 24 lattices in the genus of BW classified in [S-V]; the remaining orbits of norm 0 vectors should not be hard to find. Example 1.3. Take L to be the lattice I_{1,9} and R to be the group generated by reflections of norm −1 vectors. (This has infinite index in the full reflection group.) Then the lattice has a Weyl vector for the reflection group as in [B90], so we can apply the algorithm to this reflection group. (However it is not entirely clear what the point of doing this is, as it is easier to use the full reflection group of the lattice!)

Next we look at the second case when u^{⊥} has a root r such that v = u − r is in D. Then −r is in the fundamental domain of the finite reflection group of u^{⊥}, so r is a sum of the simple roots of u^{⊥} with the usual multiplicities.

For u in D we let R_{i}(u) be the simple roots u of D that have inner product i(r, r)/2 with u, so R_{i}(u) is empty for i < 0 and R_{0}(u) is the Dynkin diagram of u^{⊥}. We write S(u) for R_{0}(u) ∪ R_{1}(u) ∪ R_{2}(u). Then given S(v) we can find all vectors u of D that come from v as in (3) above, and S(u) is contained in S(v). By keeping track of the action of Aut(D, v) on S(v) for vectors v of D we can find all possible vectors v constructed in this way from v, together with the sets S(u).

Finally, the third case, when there is at least one root in u^{⊥}, but if r is a root in u^{⊥} then v − r is never in D, has to be dealt with separately for each lattice L. In practice it does not present too much difficulty for lattices with a vector w as in case 1. See the next section for the example of L = II_{1,25}.

The following two lemmas will be used later to prove some properties of the root systems of 25 dimensional lattices.

Lemma 1.4. Suppose that reflection in u^{⊥} is an automorphism of L. Then there is an automorphism σ of L (of order 1 or 2) with the following properties:

(1) σ fixes D.

(2) If σ fixes w, then w is a linear combination of u and the roots of L in u^{⊥}.

Proof. There is an automorphism of L acting as 1 on u and as −1 on u^{⊥}, given by the product of −1 and reflection in u^{⊥}. As this automorphism fixes u ∈ D, we can multiply it by some (unique) element of the reflection group of u^{⊥} so that the product σ fixes D. The element σ acts as −1 on the space orthogonal to u, z, and all roots of R in u^{⊥}, which implies assertion (2) of the lemma 1.4.

Lemma 1.5. Suppose that there is a norm 0 vector z such that (z, u) = 2, where u is a vector in D. Then there is an automorphism σ of L with the following properties:

(1) σ fixes D.

(2) If σ fixes w, then w is a linear combination of u, z, and the roots of L in u^{⊥}.

Proof. If M is the lattice spanned by z and u then M has the property that all elements of M′/M have order 1 or 2. So there is an automorphism of L acting as 1 on M and −1 on M^{⊥}. The result now follows as in the proof of lemma 1.4. This proves lemma 1.5.

Remark. It is usually easy to classify all orbits of negative norm vectors u in Lorentzian lattices, because this is closely related to the classification of the indefinite lattices u^{⊥}, and by Eichler’s theorem [E] indefinite lattices in dimension at least 3 are classified by the spinor genus (which in practice is often determined by the genus). For example, it is easy to give a proof along these lines that if n > 0 and m > 0 then II_{1,8n+1} has a unique orbit of primitive vectors of norm −2m.

2. Vectors in the lattice II_{1,25}.

In this section we specialize the algorithm of the previous section to the lattice II_{1,25}. Note that orbits norm 4 vectors u of II_{1,25} correspond naturally to 25 dimensional

positive definite unimodular lattices, because u^{⊥} is isomorphic to the lattice of even vectors in a 25 dimensional unimodular negative definite lattice. In particular we can classify the 665 positive definite 25 dimensional unimodular lattices, as in table 2; this is the main application of the algorithm of the previous section. Similarly norm 2 vectors of II_{1,25} correspond to 25 dimensional even positive definite lattices of determinant 2. (Another interpretation of the vectors of II_{25,1} of norm at least −2 is that they are the roots of the fake monster Lie algebra.)

First we have to show the existence of a vector w satisfying the property of section 1. This follows from Conway’s theorem [C85] stating that the reflection group of II_{1,25} has a Weyl vector w of norm 0, with the property that (w, r) = 1 for all simple roots r of the reflection group. Conway’s proof depends on the rather hard classification of the “deep holes” in the Leech lattice in [C-P-S]; there is a proof avoiding these long calculations in [B85]. It seems likely that 26 is the largest possible dimension of a lattice with a suitable vector w.

Next we have to classify the vectors u of D such that u^{⊥} has roots but u + r is not in D for any root r ∈ u^{⊥}. One obvious way this can happen is if u has norm 0, so we have to classify the norm 0 vectors in II_{1,25}. In any lattice L = II_{8n+1,1} the orbits of primitive norm 0 vectors z correspond to the 8n-dimensional even negative definite unimodular lattices z^{⊥}/z. So the orbits of primitive norm 0 vectors of II_{1,25} correspond to the 24 Niemeier lattices ([C-S]). The non-primitive norm 0 vectors are of course either 0 or a positive integer multiple of a primitive norm 0 vector, so this gives the classification of all orbits of norm 0 vectors in II_{1,25}; see table 0.

Next suppose that u is a positive norm vector of D with (u, u) = 2n and r is a highest root in u^{⊥} such that u − r is not in D. Then u − r is conjugate under the reflection group to some vector v such that (v, u) < (u − r, u). But (v, u)^{2} ≥ (u, u)(v, v) = 2n(2n − 2) and (v, u) < (u − r, u) = 2n, so (v, u) = 2n − 1. So if z = u − v then (z, u) = 1 and z^{2} = 0. If we put z′ = u − nz then z and z′ are norm 0 vectors with (z, z′) = 1 and u = nz + z′. So II_{1,25} = B ⊕ ⟨z, z′⟩ for some Niemeier lattice B. If this Niemeier lattice has roots, then adding some of these roots to r gives a vector in D by the previous argument, so B must be the Leech lattice so we can assume that z is in the orbit of w. If n > 1 then there are no roots in u^{⊥}, and if n < 1 then (u, u) ≤ 0, so we must have n = 1. So the only possibility for u is that it is a norm 2 vector in the orbit of w + w′ = 2w + r, where r is a simple root.

Putting everything together gives the following list of the vectors u ∈ D such that u^{⊥} has roots but u − r is not in D for any root r ∈ u^{⊥}:

1. The zero vector.

2. The norm 0 vectors nz for n ≥ 1 and z a primitive norm 0 vector of D corresponding to some Niemeier lattice other than the Leech lattice. The vectors for a given Niemeier lattice and a given value of n are all conjugate under Aut(D).

3. The norm 2 vectors of the form 2w + r for a simple root r of D. These form one orbit under Aut(D).

Lemma 2.1. Suppose u, v ∈ D, u^{2} = 2n, v^{2} = 2(n − 1) and (v, u) = 2n. Then

\begin{aligned}R_{0}(u) & ⊆ R_{0}(v) ∪ R_{1}(v) ∪ R_{2}(v) = S(v) \\ R_{i}(u) & ⊆ R_{0}(v) ∪ R_{1}(v) ∪ ⋯ ∪ R_{i}(v) for i ≥ 1. \end{aligned}

Proof. The vector v is in D, so v = u + r for some highest root r of u^{⊥}. The vector r has inner product 0, 1, or 2 with all simple roots of u^{⊥}, and −r is a sum of roots of R_{0}(u) with positive coefficients, so r has inner product ≥ 0 with all simple roots of D not in R_{0}(u). The lemma follows from this and the fact that (v, s) = (u, s) + (r, s) for any simple root s of D. This proves lemma 2.1.

We now start with a vector v of norm 2(n − 1) and try to reconstruct u from it. The vector u − v is a highest root of some component of R_{0}(u), and R_{0}(u) is contained in S(v), so we should be able to find u from S(v). By lemma 2.1 S(u) is contained in S(v), so we can repeat this process with u instead of v. The following theorem shows how to construct all possible vectors u as in lemma 2.1 from v and S(v).

Theorem 2.2. Suppose that v has norm 2(n − 1) and is in D (so n ≥ 1). Then there are bijections between

(1) Norm 2n vectors u of D with (u, v) = 2n.

(2) Simple spherical Dynkin diagrams C contained in the Dynkin diagram Λ of D such that if r is the highest root of C and c in C satisfies (c, r) = i, then c is in R_{i}(v).

(3) Dynkin diagrams C satisfying one of the following three conditions: Either C is an a_{1} and is contained in R_{2}(v),

or C is an a_{n} (n ≥ 2) and the two endpoints of C are in R_{1}(v) while the other points

of C are in R_{0}(v), or C is d_{n} (n ≥ 4), e_{6}, e_{7}, or e_{8} and the unique point of C that has inner product 1 with the highest root of C is in R_{1}(v) while the other points of C are in R_{0}(v).

Proof. Let u be as in (1) and put r = u − v. The vector r is orthogonal to u and has inner product ≤ 0 with all roots of R_{0} (because −v does) so it is a highest root of some component C of R_{0}(u). The vector r therefore determines some simple spherical Dynkin diagram C contained in Λ. Any root c of C has (c, v + r) = (c, u) = 0, so c is in R_{i}(v) where i = (c, r). This gives a map from (1) to (2).

Conversely if we start with a Dynkin diagram C satisfying (2) and put u = v + r (where r is the highest root of C) then (c, u) = 0 for all c in C, so (r, u) = 0 as r is a sum of the c’s. This implies that u^{2} = 2n and (u, v) = 2n. We now have to show that u is in D. Let s be any simple root of D. If s is in C then (s, r) = −(s, v) and if s is not in C then (s, r) ≥ 0, so in any case (s, u) = (s, v + r) ≥ 0 and hence u is in D. This gives a map from (2) to (1) and shows that (1) and (2) are equivalent.

Condition (3) is just the condition (2) written out explicitly for each possible C, so (2) and (3) are also equivalent. This proves theorem 2.2.

We define the height of a vector u in II_{1,25} to be (u, w). We show how to calculate the heights of vectors of II_{1,25} that have been found with the algorithm above.

Lemma 2.3. Suppose u, v are vectors in D of norms 2n, 2(n − 1) with (u, v) = 2n and suppose that v = u − r for some root r of u^{⊥} corresponds to the component C of R_{0}(u). Then

height(u) = height(v) + h − 1

where h is the Coxeter number of the component C.

Proof. We have v = u − r where r is the highest root of C, so height(u) = height(v) + (r, w). We have r = ∑_{i} m_{i}c_{i} where the c_{i} are the simple roots of C with weights m_{i} and ∑_{i} m_{i} = h − 1. All the c_{i} have inner product 1 with w, so (r, w) = h − 1. This proves lemma 2.3.

Lemma 2.4. Let u be a primitive vector of D such that there is a norm 0 vector z with (z, u) = 0 or 1, and suppose that z corresponds to a Niemeier lattice B with Coxeter number h.

(1) If u has norm 0 then its height is h. The Dynkin diagram of u^{⊥} is the extended Dynkin diagram of B.

(2) If u has positive norm then height(u) = 1+(1+u^{2}/2)h. The Dynkin diagram of u^{⊥} is the Dynkin diagram of B if u^{2} > 2 and the Dynkin diagram of B plus an a_{1} if u^{2} = 2.

Proof.

(1) The Dynkin diagram of u^{⊥} is a union of extended Dynkin diagrams. If this union is empty then u must be w and therefore has height 0 = h. If not then let C be one of the components. We have u = ∑_{i} m_{i}c_{i} where the c_{i}’s are the simple roots of C with weights m_{i}. Also ∑_{i} m_{i} = h because C is an extended Dynkin diagram and all the c_{i}’s have height 1, so u has height h.

(2) As u has inner product 1 with a norm 0 vector z of D we can put u = nz + z′ with u^{2} = 2n and z^{′2} = 0, (z, z′) = 1. By part (1) z has height h. We have z′ = z + r where r is a simple root of D, so height(z′) = height(z) + height(r) = h + 1. Hence height(u) = nh + h + 1 = 1 + (1 + u^{2}/2)h. The lattice u^{⊥} is B ⊕ N where N is a one dimensional lattice of determinant 2n, so the Dynkin diagram is that of B plus that of N, and the Dynkin diagram of (norm 2 roots of) N is empty unless 2n = 2 in which case it is a_{1}. This proves that the Dynkin diagram of u^{⊥} is what it is stated to be. This proves lemma 2.4. Orbits of norm 2 vectors u ∈ II_{1,25} correspond to even 25 dimensional positive definite

lattices B of determinant 2, where B(−1) ≅ u^{⊥}. One part of the algorithm for finding vectors of norm 2n consists of finding the vectors u such that there are no roots in u^{⊥}. For norm 2 vectors u the following lemma shows that there are no such vectors.

Lemma 2.5. If u ∈ II_{1,25} has norm 2 then u^{⊥} contains roots. In other words every 25 dimensional even positive definite lattice of determinant 2 has a root.

Proof. If u^{⊥} contains no roots then, by the algorithm of section 1, u = w + u_{1} for some u_{1} in D. We have u_{1}^{2} = u^{2} −2height(u), so u_{1}^{2} = 0 and u has height 1 because u_{1}^{2} ≥ 0, u^{2} = 2 and the height of u is positive. Then height(u_{1}) = height(u) = 1, so u is a norm 0 vector in D that has inner product 1 with the norm 0 vector w of D, but this is impossible as u − w would be a norm −2 vector separating the two vectors u and w of D. This proves lemma 2.5.

Theorem 2.6. Suppose that u ∈ D has norm 2. Then

w = ρ + height(u)u/2

where ρ is the Weyl vector of the root system of u^{⊥}. Also −2ρ^{2} = height(u)^{2}.

Proof. The vector w is fixed by any automorphism fixing D, so by lemma 1.4 the vector w must be in the space spanned by u and the roots of u^{⊥}. However w also has inner product 1 with all simple roots of u^{⊥} and has inner product height(u) with u, so w must be ρ + height(u)u/2. Taking norms of both sides of w = ρ + height(u)u/2, and using the facts that w^{2} = 0, (u, ρ) = 0, and (u, u) = 2, shows that −2ρ^{2} = height(u)^{2}. This proves theorem 2.6.

In particular we find the strange consequence that the norm of the Weyl vector of any 25 dimensional even positive definite lattice of determinant 2 must be a half a square.

Norm 4 vectors in the fundamental domain D of II_{1,25} correspond to 25 dimensional unimodular lattices A = A_{1} ⊕ I^{n}, where u^{⊥} is the lattice of even elements of A(−1) and A_{1} has no norm 1 vectors . The odd vectors of A(−1) can be taken as the projections of the vectors y with (y, u) = 2 into u^{⊥}. A norm 4 vector u can behave in 4 different ways, depending on whether the unimodular lattice A_{1} with no norm 1 vectors corresponding to u is at most 23 dimensional, or 24 dimensional and odd, or 24 dimensional and even, or 25 dimensional.

Theorem 2.7. Norm 1 vectors of A correspond to norm 0 vectors z of II_{1,25} with (z, u) = 2. Write A = A_{1} ⊕ I^{n} where A_{1} has no vectors of norm 1. Then u is in exactly one of the following four classes:

(1) u has inner product 1 with a norm 0 vector. The lattice A_{1} is a Niemeier lattice.

(2) A has at least 4 vectors of norm 1, so that A_{1} is at most 23 dimensional (but may be even). There is a unique norm 0 vector z of D with (z, u) = 2 and this vector z is of the same type as either of the two even neighbors of A_{1} ⊕ I^{n−1}.

(3) A_{1} is 24 dimensional and odd. There are exactly two norm 0 vectors that have inner product 2 with u, and they are both in D. They have the types of the two even neighbors of A_{1}.

(4) A = A_{1} has no vectors of norm 1.

Proof. The vector z is a norm 0 vector with (z, u) = 2 if and only if u/2 − z is a norm 1 vector of A. Most of 2.7 follows from this. The only non-trivial things to check are the statements about norm 0 vectors that are in D.

If u does not have inner product 1 with any norm 0 vector then a norm 0 vector z with (z, u) = 2 is in D if and only if it has inner product ≥ 0 with all simple roots of u^{⊥}, so there is one such vector in D for each orbit of such norm 0 vectors under the reflection group of u^{⊥}. If A has at least 4 vectors of norm 1 then they form a single orbit under the Weyl group of (the norm 2 vectors of) u^{⊥}, which proves (2), while if A has only two vectors of norm 1 then they are both orthogonal to all norm 2 vectors of A and so form two orbits under they Weyl group of u^{⊥}. This proves theorem 2.7.

Theorem 2.8. Suppose that u is a norm 4 vector corresponding to a unimodular 25 dimensional lattice A = A_{1} ⊕ I^{25−n} with 2n ≥ 4 vectors of norm 1. Let ρ be the Weyl vector of the root system of norm −2 roots of u^{⊥} (which is the Weyl vector of the norm −2 vectors of A(−1)) and let let h be the Coxeter number of the even neighbors of the 24 dimensional unimodular lattice A_{1} ⊕ I^{24−n}. Then height(u) = (w, u) = 2(h + n − 1), w = ρ + height(u)u/4, and −ρ^{2} = (h + n − 1)^{2}.

Proof. There is a unique norm 2 vector z of D with (z, u) = 2; we let i be its projection into u^{⊥}. The lattice A has at least 4 vectors of norm 1, so any vector of norm 1 and in particular i is in the vector space generated by vectors of norm −2 of u^{⊥}. Hence by lemma 1.5 and the same argument as in theorem 2.6 we have w = ρ+height(u)u/4. The norm −4 vector 2i of u^{⊥} is the sum of −2(n − 1) simple roots of the d_{n} component of the Dynkin diagram of u^{⊥}, so (2i, w) = (2i, ρ) = −2(n − 1).

The vector i is the projection of z into u^{⊥}, so i = z − u/2, and hence

height(u) = (w, u)
= 2(w, z − i)
= 2(height(z) + n − 1)
= 2(h + n − 1).

If we calculate the norms of both sides of w = ρ + height(u)u/4 we find that −ρ^{2} = (h + n − 1)^{2}. This proves theorem 2.8. Example 2.9. Suppose u corresponds to the lattice I^{25}. The number n is then 25 and the root system of the norm 2 vectors is D_{25}, so the Weyl vector ρ can be taken as (0, 1, 2, …, 24). The even neighbors of I^{24} are both D_{24} with Coxeter number h = 46, so we find that 0^{2} + 1^{2} + 2^{2} + ⋯ + 24^{2} = ρ^{2} = (h + n − 1)^{2} = 70^{2}. Watson [W] showed that the only solution of 0^{2} + 1^{2} + ⋯ + k^{2} = m^{2} with k ≥ 2 is k = 24. See [C-S Chapter 26] for a construction of the Leech lattice using this equality.

Theorem 2.10. Suppose that u is a norm 4 vector of D with exactly two norm 0 vectors z_{1}, z_{2} that have inner product 2 with u, and suppose that there are no norm 0 vectors that have inner product 1 with u. Then z_{1} and z_{2} are both in D and have Coxeter numbers h_{1}, h_{2} where h_{i} = (z_{i}, w). Then

w = ρ + (h_{1}z_{2} + h_{2}z_{1})/2

where ρ is the Weyl vector of the norm −2 vectors of u^{⊥}. Also u = z_{1} + z_{2}, height(u) = h_{1} + h_{2}, −ρ^{2} = h_{1}h_{2}, and u^{⊥} has 8(h_{1} + h_{2} − 2) roots.

Proof. The vector u − z_{1} is a norm 0 vector which has inner product 2 with u and so must be z_{2}. Hence u = z_{1} + z_{2} and height(u) = height(z_{1}) + height(z_{2}) = h_{1} + h_{2}.

There is a norm 0 vector that has inner product 2 with u, and any automorphism of L fixing D also fixes w, so by lemma 1.5 w is a linear combination of z_{1}, z_{2}, and the roots of R in u^{⊥}. Using the facts that (w, z_{1}) = h_{1}, (w, z_{2}) = h_{2}, and (w, r) = −r^{2}/2 for any simple root r in u^{⊥} shows that w must then be ρ + (h_{1}z_{2} + h_{2}z_{1})/2. Using the fact that w^{2} = 0 this shows immediately that −ρ^{2} = h_{1}h_{2}. The number of roots follows from remark 2.12 below. This proves theorem 2.10.

Corollary 2.11. If A_{1} is an odd 24 dimensional positive definite unimodular lattice with no vectors of norm 1 and whose even neighbors have Coxeter numbers h_{1} and h_{2}, then ρ^{2} = h_{1}h_{2} where ρ is the Weyl vector of A_{1}.

Proof. This follows immediately from theorem 2.10, using the fact that A_{1} ⊕ I is the 25 dimensional unimodular lattice corresponding to u as in 2.10.

Remark. Let B_{1}, B_{2} be the two even neighbors of A_{1}. Then it is not hard to show that h_{2} ≤ 2h_{1} + 2, and there are several lattices A_{1} for which equality holds. Remark 2.12. Theorem 13.1 and corollary 13.2 of [B95] show that the height of a vector in the fundamental domain of II_{1,25} can be written as an explicit linear combination of the theta functions of cosets of the lattice u^{⊥}. In particular we find that if u is a norm 2 vector then

12height(u) = 18 − 4z_{1} + r

where r is the number of norm −2 vectors of u^{⊥} and z_{i} is the number of norm 0 vectors having inner product i with u (so z_{1} is 0 or 2 and is 2 if and only if the lattice u^{⊥} is the sum of a one dimensional lattice and an even lattice). Similarly if u has norm 4 and corresponds to a 25 dimensional unimodular lattice A then

8t = 20 − 2z_{2} − 8z_{1} + r

where r is the number of norm 2 vectors of A, z_{2} is the number of norm 1 vectors of A, and z_{1} is 1 if A is the sum of a Niemeier lattice and a one dimensional lattice and is 0 otherwise. Note that these relations give congruences for the numbers of roots that immediately imply that 25 dimensional even lattices of determinant 2 and 25 dimensional unimodular lattices always have roots. There are similar relations and congruences for larger norm vectors of II_{1,25}.

There are several other genuses of lattices that can be classified using II_{1,25}. Most of these do not seem important enough to be worth publishing, but here is a summary of what is available just in case anyone finds a use for any of these. The 24 dimensional even positive definite lattices of determinant 5 are easy to classify as they turn out to correspond to pairs consisting of a norm 2 vector u of II_{1,25} together with a norm −2 root r with (r, u) = 1, and these can easily be read off from the list of norm 2 vectors. The 25 dimensional positive definite even lattices of determinant 6 correspond to the norm 6 vectors in II_{1,25} and can be classified from the norm 4 vectors using the algorithm; there are 2825 orbits if I have made no mistakes. A list of them is available from my home page. These can be used to classify the 26 dimensional even positive definite lattices of determinant 3, because the norm 2 roots of such lattices correspond to the norm 6 vectors of II_{1,25}. (There is a unique such lattice with no roots; see the next section.) There are between 677 and 681 such lattices, and a provisional list is available from my home page (there are a few small ambiguities that I have not yet got around to resolving). If such a lattice has no norm 6 roots then the number of norm 2 vectors is divisible by 6. With a lot more effort it should be possible to classify the 26 dimensional unimodular lattices by finding the (roughly 50000?) orbits of norm 10 vectors of II_{1,25}; see the next section.

3. Lattices with no roots.

In this section we show that there is a unique 26 dimensional positive definite unimodular lattice with no roots. Conway and Sloane use this result in their proof [C-S98] that there is a positive definite unimodular lattice with no roots in all dimensions greater than 25. We also show that the number of norm 2 vectors of a 26 dimensional unimodular lattice is divisible by 4, and sketch a construction of a 27 dimensional unimodular lattice with no roots.

Lemma 3.1. A 26-dimensional unimodular lattice L with no vectors of norm 1 has a characteristic vector of norm 10.

Proof. If L has a characteristic vector x of norm 2 then x^{⊥} is a 25 dimensional even lattice of determinant 2 and therefore has a root r by theorem 2.6; 2r +x is a characteristic vector of norm 10. If the lemma is not true we can therefore assume that L has no vectors of norm 1 and no characteristic vectors of norm 2 or 10. Its theta function is determined by these conditions and turns out to be 1−156q^{2} +⋯ which is impossible as the coefficient of q^{2} is negative. This proves lemma 3.1.

Lemma 3.2. There is a bijection between isomorphism classes of

(1) Norm 10 characteristic vectors c in 26-dimensional positive definite unimodular lattices L, and

(2) Norm 10 vectors u in II_{1,25} given by c^{⊥}(−1) ≅ u^{⊥}. We have Aut(L, c) = Aut(II_{1,25}, u).

Proof. Routine. Note that −1 is a square mod 10. This proves lemma 3.2.

Lemmas 3.1 and 3.2 give an algorithm for finding 26 dimensional unimodular lattices L. It is probably not hard to implement this on a computer if one is given a computer algorithm for deciding when 2 vectors of the Leech lattice are conjugate under its automorphism group; such an algorithm has been described by Allcock in [A]. The main remaining open problem is to find a use for these lattices! We now apply this algorithm to find the unique such lattice with no roots.

Lemma 3.3. Take notation as in lemma 3.2. The lattice L has no roots if and only if u^{⊥} has no roots and u does not have inner product 1, 2, 3, or 4 with any norm 0 vector.

Proof. If u^{⊥} has roots then obviously L has too. If there is a norm 0 vector z that has inner product 1, 2, 3, or 4 with u then the projection z_{u} of z into u^{⊥} has norm −1/10, −4/10, −9/10, or −16/10. The lattice L(−1) contains u^{⊥} + c, and the vector z_{u} ± 3c/10, z_{u} ± 4c/10, z_{u} ± c/10, or z_{u} ± 2c/10 is in L for some choice of sign and has norm −1, −2, −1, or −2. Hence if u has inner product 1, 2, 3, or 4 with some norm 0 vector then L has roots. Conversely if L has a root r then either r has norm 2 and inner product 0, ±2, ±4 with c or it has norm 1 and inner product ±1, ±3 with c, and each of these cases implies that u^{⊥} has roots or that u has inner product 1, 2, 3, or 4 with some norm 0 vector by reversing the argument above. This proves lemma 3.3.

Now let L be a 26 dimensional unimodular lattice with no roots containing a characteristic vector c of norm 10, and let u be a norm 10 vector of D corresponding to it as in 3.2.

Lemma 3.4. u = z + w, where z is a norm 0 vector of D corresponding to a Niemeier lattice with root system A_{4}^{6}, and w is the Weyl vector of D. In particular u is determined up to conjugacy under Aut(D).

Proof. The lattice u^{⊥} has no roots so u = w +z for some vector z of D. By lemma 3.3 u does not have inner product 1, 2, 3, or 4 with any norm 0 vector, so (z, w) = (u, w) ≥ 5. Hence

10 = u^{2} = z^{2} + 2(z, w) ≥ 2(z, w) ≥ 10

so (z, w) = 5 and z^{2} = 0. The only norm 0 vectors z in D with (z, w) = 5 are the primitive ones corresponding to A_{4}^{6} Niemeier lattices, which form one orbit under Aut(D). This proves lemma 3.4.

Lemma 3.5. If u = z + w is as in lemma 3.4 then the 26 dimensional unimodular lattice corresponding to u has no roots.

Proof. The lattice u^{⊥} obviously has no roots so by lemma 3.3 we have to check that there are no norm 0 vectors that have inner product 1, 2, 3, or 4 with u. Let x be any norm 0 vector in the positive cone. If x has type A_{4}^{6} then (x, u) ≥ (x, w) ≥ 5; if x has Leech type then (x, u) ≥ (x, z) ≥ 5; if x has type A_{1}^{24} then (x, u) = (x, w) + (x, z) ≥ 2 + 3 = 5 ((x, z) cannot be 2 as there are no pairs of norm 0 vectors of types A_{1}^{24} and A_{4}^{6} that have inner product 2 by the classification of 24 dimensional unimodular lattices); and if x has any other type then (x, u) = (x, w) + (x, z) ≥ 3 + 2 = 5. This proves lemma 3.5.

Theorem 3.6. There is a unique 26 dimensional positive definite unimodular lattice L with no roots. Its automorphism group is isomorphic to the group O_{5}(5) = 2.G.2 of order 2^{8}.3^{2}.5^{4}.13 and acts transitively on the 624 characteristic norm 10 vectors of L.

Proof. By lemma 3.1 L has a characteristic vector of norm 10, so by lemmas 3.3 and 3.4 L is unique and its automorphism group acts transitively on the characteristic vectors of norm 10. By lemma 3.5 L exists. The theta function is determined by the conditions that L has no vectors of norm 1 or 2 and no characteristic vectors of norm 2, and it turns out that the number of characteristic vectors of norm 10 is 624. The stabilizer of such a vector is isomorphic to Aut(II_{1,25}, u), which is a group of the form 5^{3}.2.S_{5} where S_{5} is the symmetric group on 5 letters. This determines the order of the automorphism group of the lattice. From this it is not difficult to determine it precisely; we omit the details. This proves theorem 3.6.

We now show that the number of norm 2 vectors of any 26 dimensional even positive definite unimodular lattice is divisible by 4. There are strictly 26 dimensional unimodular lattices with no roots or with 4 roots, so this is the best possible congruence. For unimodular lattices of dimension less than 26 there are congruences modulo higher powers of 2 for the number of roots.

Lemma 3.7. If L is a 25-dimensional positive definite lattice of determinant 2 then the number of norm 2 roots of L is 2 mod 4.

Proof. The even vectors of L form a lattice isomorphic to the vectors that have even inner product with some vector b in an even 25-dimensional lattice B of determinant 2. (Note that b is not in B′ − B.) The number of roots of B is 12t − 10 or 12t − 18 where t is the height of the norm 2 vector of D corresponding to B by remark 2.12, so it is sufficient to prove that the number of norm 2 vectors of B that have odd inner product with b is divisible by 4.

The vector b has zero inner product with u and integral inner product with w, so by theorem 2.6 b has integral inner product with ρ. Hence b has even inner product with the sum of the positive roots of B, so it has odd inner product with an even number of positive roots. This implies that the number of roots of B that have odd inner product with b is divisible by 4. This proves lemma 3.7.

Corollary 3.8. If L is a 26 dimensional unimodular lattice then the number of norm 2 vectors of L is divisible by 4.

Proof. The result is obvious if L has no norm 2 roots, so let r be a norm 2 vector of L. The lattice r^{⊥} is a 25 dimensional even lattice of determinant so by remark 2.12 the number of roots of r^{⊥} is 2 mod 4. The number of roots of L not in r^{⊥} is 4h − 6 where h is the Coxeter number of the component of L containing r, so the number of norm 2 vectors of L is divisible by 4. This proves corollary 3.8.

Remark. A similar but more complicated argument can be used to show that there is a unique even 26 dimensional positive definite lattice of determinant 3 with no roots. Gluing on a one dimensional lattice to this gives a unique 27 dimensional unimodular lattice with no roots and a characteristic vector of norm 3. As a different proof of this has already been published in [E-Z] we will just give a brief sketch of the proof from [B]. (The preprint [B-V] shows that there are exactly three 27 dimensional positive definite unimodular lattices with no roots.) Let L be a 27 dimensional positive definite unimodular lattice with no roots and a characteristic vector c of norm 3. The theta function of L is determined by these conditions and this implies that L has vectors of norm 5; let v be such a vector. Then ⟨v, c⟩^{⊥} is a 25 dimensional even lattice X of determinant 14 such that X′/X is generated by an element of norm 1/14 mod 2. Such lattices X correspond to norm 14 vectors x in the fundamental domain D of II_{1,25}, and the condition that L has no vectors of norm 1 or 2 implies that there are exactly two possibilities for x: x is either the sum of w and a norm 0 vector of height 7 corresponding to A_{6}^{4}, or x is the sum of w and a norm 2 vector of height 6 corresponding to the 25 dimensional lattice of determinant 2 with root system a_{2}^{9}. Both of these x’s turn out to give the same lattice L, which therefore has two orbits of norm 5 vectors and is the unique 27 dimensional positive definite unimodular lattice with no roots and with characteristic vectors of norm 3.

Table 0. The primitive norm 0 vectors of II_{1,25}.

We list the set of orbits of primitive norm 0 vectors z of II_{1,25}, which is of course more or less the same as the well known list of Niemeier lattices (see [C-S table 16.1]). The height is just (w, z) where w is the Weyl vector of a fundamental domain containing z. 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 0 vectors” of table 1. The column headed “Group” is the order of the subgroup of Aut(D) fixing the primitive norm 0 vector. However note that the group order is not (usually) the order of the quotient of the automorphism group of the Niemeier lattice by the reflection group; see [C-S chapter 16] for a description of the relation between these groups. For the vector w of height 0 the group is the infinite group of automorphisms of the affine Leech lattice and is an extension of a finite group of the order given by the group of translations of the Leech lattice Λ.

Height  Roots  Group
0x  None Λ·8315553613086720000
2a  A_{1}^{24}  1002795171840
3a  A_{2}^{12}  138568320
4a  A_{3}^{8}  688128
5a  A_{4}^{6}  30000
6d  D_{4}^{6}  138240

6a  A_{5}^{4}D_{4}  3456
7a  A_{6}^{4}  1176
8a  A_{7}^{2}D_{5}^{2}  256
9a  A_{8}^{3}  324
10d  D_{6}^{4}  384
10a  A_{9}^{2}D_{6}  80
12e  E_{6}^{4}  432
12a A_{11}D_{7}E_{6}  24
13a  A_{12}^{2}  52
14d  D_{8}^{3}  48
16a A_{15}D_{9}  16
18d D_{10}E_{7}^{2}  8
18a A_{17}E_{7}  12
22d  D_{12}^{2}  8
25a  A_{24}  10
30e  E_{8}^{3}  6
30d D_{16}E_{8}  2
46d  D_{24}  2

Table 1. The norm 2 vectors of II_{1,25}.

The following sets are in natural 1:1 correspondence:

(1) Orbits of norm 2 vectors in II_{1,25} under Aut(II_{1,25}).

(2) Orbits of norm 2 vectors u of D under Aut(D).

(3) 25 dimensional even bimodular lattices L.

The lattice L(−1) is isomorphic to u^{⊥}. Table 1 lists the 121 elements of any of these three sets.

The height 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” of table 2. 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 a_{1}.

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” describes the norm 0 vectors z corresponding to each orbit of roots of u^{⊥} where u is in D. 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 E_{6}^{4}, A_{11}D_{7}E_{6}, and D_{10}E_{7}^{2}.

For some remarks on the reliability of table 1 see the introduction to table 2.

Height  Roots  Group S Norm 0 vectors
1a*  a_{1} 8315553613086720000 1  X
2a  a_{2}  991533312000 1  x
3a  a_{1}^{9}  92897280 9  a
4a  a_{2}a_{1}^{12}  190080 13  aa
5a*  a_{1}^{24}a_{1}  244823040 25  aA
5b  a_{2}^{4}a_{1}^{9}  3456 13  aa
5c  a_{3}a_{1}^{15}  40320 17  aa
6a  a_{2}^{9}  3024 9  a
6b  a_{3}a_{2}^{5}a_{1}^{6}  240 13  aaa
7a*  a_{2}^{12}a_{1}  190080 13  aA
7b  a_{3}^{3}a_{2}^{4}a_{1}^{3}  48 13  aaa
7c  a_{3}^{4}a_{1}^{8}a_{1}  384 17  aad
7d  a_{4}a_{2}^{6}a_{1}^{5}  240 13  aaa
7e  d_{4}a_{1}^{21}  120960 25  ad
8a  a_{3}^{6}a_{2}  240 13  ad
8b a_{4}a_{3}^{3}a_{2}^{3}a_{1}^{2}  12 13 aaaa
8c  d_{4}a_{2}^{9}  864 13  aa
9a*  a_{3}^{8}a_{1}  2688 17  aA
9b  a_{4}^{2}a_{3}^{4}a_{1}  16 13  aaa
9c a_{4}^{3}a_{3}a_{2}^{2}a_{1}^{3}  12 13 aaaa
9d d_{4}a_{3}^{4}a_{3}a_{1}^{3}  48 17 aada
9e  a_{5}a_{3}^{3}a_{2}^{4}  24 13  aaa
9f  a_{5}a_{3}^{4}a_{1}^{6}  48 17  aaa
10a  d_{4}a_{4}^{3}a_{2}^{3}  12 13  aaa
10b a_{5}a_{4}^{2}a_{3}^{2}a_{2}a_{1}  4 13 aaaaa
11a*  a_{4}^{6}a_{1}  240 13  aA
11b  d_{4}^{4}a_{1}^{9}  432 25  dd
11c  a_{5}d_{4}^{2}a_{3}^{3}  24 17  daa
11d a_{5}a_{5}a_{4}^{2}a_{3}a_{1}  4 13 aaaaa

11e a_{5}^{2}d_{4}a_{3}^{2}a_{1}^{2}a_{1}  8 17 aaaad
11f  a_{5}^{3}a_{2}^{4}  48 13  aa
11g  d_{5}a_{3}^{6}a_{1}  48 17  aad
11h a_{6}a_{4}^{2}a_{3}^{2}a_{2}a_{1}  4 13 aaaaa
12a  a_{5}^{4}a_{2}  24 13  ad
12b  d_{5}a_{4}^{4}a_{2}  8 13  aaa
12c  a_{6}d_{4}a_{4}^{3}  6 13  aaa
12d a_{6}a_{5}^{2}a_{3}a_{2}^{2}  4 13 aaaa
13a*  a_{5}^{4}d_{4}a_{1}  48 17 aaA
13b d_{5}a_{5}^{2}d_{4}a_{3}a_{1}  4 17 aaada
13c d_{5}a_{5}^{3}a_{1}^{3}a_{1}  12 17 aaae
13d* d_{4}^{6}a_{1} 2160 25 dD
13e a_{6}^{2}a_{5}a_{4}a_{1}^{2}  4 13 aaaa
13f a_{7}a_{5}a_{4}^{2}a_{3} 4 13 aaaa
13g a_{7}a_{5}d_{4}a_{3}^{2}a_{1}^{2}  4 17 aaaaa
14a a_{6}a_{6}d_{5}a_{4}a_{2}  2 13 aaaaa
14b a_{6}^{3}d_{4} 12 13 aa
14c a_{7}a_{6}a_{5}a_{4}a_{1}  2 13 aaaaa
15a*  a_{6}^{4}a_{1}  24 13  aA
15b  d_{5}^{3}a_{5}a_{3}  12 17  ade
15c  d_{6}d_{4}^{4}a_{1}^{3}  24 25 ddd
15d d_{6}a_{5}^{2}a_{5}a_{3}  4 17 aada
15e a_{7}d_{5}^{2}a_{3}^{2}a_{1}  4 17 aaad
15f  a_{7}^{2}d_{4}^{2}a_{1}  8 17  aad
15g  a_{8}a_{5}^{3}  6 13  aa
15h a_{8}a_{6}a_{5}a_{3}a_{2}  2 13 aaaaa
16a  a_{7}^{3}a_{2}  12 13  ad
16b a_{8}a_{6}d_{5}a_{4}  2 13 aaaa
17a*  a_{7}^{2}d_{5}^{2}a_{1}  8 17 aaA
17b  e_{6}a_{5}^{3}d_{4}  12 17  aae
17c a_{7}d_{6}d_{5}a_{5}  2 17 daaa
17d a_{7}^{2}d_{6}a_{3}a_{1}  4 17 aada
17e a_{8}a_{7}^{2}a_{1} 4 13 aaa
17f a_{9}d_{5}a_{5}d_{4}a_{1}  2 17 aaaaa
17g  a_{9}a_{7}a_{4}^{2}  4 13  aaa
18a  e_{6}a_{6}^{3}  6 13  aa
18b a_{9}a_{8}a_{5}a_{2}  2 13 aaaa

19a*  a_{8}^{3}a_{1}  12 13  aA
19b  d_{6}^{3}d_{4}a_{1}^{3}  6 25 ddd
19c a_{7}e_{6}d_{5}^{2}a_{1}  4 17 eaad
19d d_{7}a_{7}d_{5}a_{5}  2 17 aaad
19e d_{7}a_{7}^{2}a_{3}a_{1}  4 17 aaad
19f a_{9}a_{7}d_{6}a_{1}a_{1}  2 17 aaaad
19g a_{10}a_{7}a_{6}a_{1}  2 13 aaaa
20a  a_{8}^{2}e_{6}a_{2}  4 13  aaa
20b  a_{10}a_{8}d_{5}  2 13  aaa
21a*  a_{9}^{2}d_{6}a_{1}  4 17 aaA
21b a_{11}d_{6}a_{5}a_{3}  2 17 aaaa
21c  a_{11}a_{8}a_{5}  2 13  aaa
21d*  d_{6}^{4}a_{1}  24 25  dD
21e a_{9}e_{6}d_{6}a_{3}  2 17 aaad
23a  d_{7}e_{6}^{2}a_{5}  4 17  ead
23b d_{8}d_{6}^{2}d_{4}a_{1}  2 25 dddd
23c  a_{9}d_{7}^{2}  4 17  da
23d  a_{9}d_{8}a_{7}  2 17  daa
23e a_{11}d_{7}d_{5}a_{1}  2 17 aaad
24a  a_{11}^{2}a_{2}  4 13  ad
24b  a_{12}e_{6}a_{6}  2 13  aaa
25a* a_{11}d_{7}e_{6}a_{1}  2 17 aaaA
25b  a_{13}d_{6}d_{5}  2 17  aaa
25e*  e_{6}^{4}a_{1}  48 17  eE
26a  a_{13}a_{10}a_{1}  2 13  aaa
27a*  a_{12}^{2}a_{1}  4 13  aA
27b  e_{7}d_{6}^{3}  3 25  dd
27c  a_{9}a_{9}e_{7}  2 17  ada
27d  d_{9}a_{9}e_{6}  2 17  ada
27e  a_{11}d_{9}a_{5}  2 17  aad
27f  a_{14}a_{9}a_{2}  2 13  aaa
29a  a_{11}e_{7}e_{6}  2 17  daa
29d*  d_{8}^{3}a_{1}  6 25  dD
31a d_{8}^{2}e_{7}a_{1}a_{1}  2 25 ddde

31b d_{10}d_{8}d_{6}a_{1}  1 25 dddd
31c  a_{15}d_{8}a_{1}  2 17  aad
33a* a_{15}d_{9}a_{1}  2 17 aaA
33b  a_{15}e_{7}a_{3}  2 17  aad
33c  a_{17}a_{8}  2 13  aa
35a  e_{7}^{3}d_{4}  6 25  de
35b  a_{13}d_{11}  2 17  da
36a  a_{18}e_{6}  2 13  aa
37a*  a_{17}e_{7}a_{1}  2 17 aaA
37d* d_{10}e_{7}^{2}a_{1}  2 25 ddD
39a  d_{12}e_{7}d_{6}  1 25 ddd
45d*  d_{12}^{2}a_{1}  2 25  dD
47a  d_{10}e_{8}e_{7}  1 25  edd
47b  d_{14}d_{10}a_{1}  1 25 ddd
47c  a_{17}e_{8}  2 17  da
48a  a_{23}a_{2}  2 13  ad
51a*  a_{24}a_{1}  2 13  aA
61d* d_{16}e_{8}a_{1}  1 25 ddD
61e*  e_{8}^{3}a_{1}  6 25  eE
63a  d_{18}e_{7}  1 25  dd
93d*  d_{24}a_{1}  1 25  dD

Table 2. The norm 4 vectors of II_{1,25}.

There is a natural 1:1 correspondence between the elements of the following sets:

(1) Orbits of norm 4 vectors u in II_{1,25} under Aut(II_{1,25}).

(2) Orbits of norm 4 vectors in the fundamental domain D of II_{1,25} under Aut(D).

(3) Orbits of norm 1 vectors v of I_{1,25} under Aut(I_{1,25}).

(4) 25 dimensional unimodular positive definite lattices L.

(5) Unimodular lattices L_{1} of dimension at most 25 with no vectors of norm 1.

(6) 25 dimensional even lattices L_{2} of determinant 4.

L_{1} is the orthogonal complement of the norm 1 vectors of L, L_{2} is the lattice of elements of L of even norm, L_{2}(−1) is isomorphic to u^{⊥}, and L(−1) is isomorphic to v^{⊥}. Table 2 lists the 665 elements of any of these sets.

The height is the height of the norm 4 vector u of D, in other words (u, w) where w is the Weyl vector of D. The things in table 2 are listed in increasing order of their height.

Dim is the dimension of the lattice L_{1}. A capital E after the dimension means that L_{1} is even.

The column “roots” gives the Dynkin diagram of the norm 2 vectors of L_{2} arranged into orbits under Aut(L_{2}).

“Group” gives the order of the subgroup of Aut(D) fixing u. The group Aut(L) ≅ Aut(L_{2}) is of the form 2 × R.G where R is the group generated by the reflections of norm 2 vectors of L, G is the group described in the column “group”, and 2 is the group of order 2 generated by −1. If dim(L_{1}) ≤ 24 then Aut(L_{1}) is of the form R.G where R is the reflection group of L_{1} and G is as above.

For any root r of u^{⊥} the vector v = u − r is a norm 2 vector of II_{1,25}. This vector v can be found as follows. Let X be the component of the Dynkin diagram of u^{⊥} to which u belongs and let h be the Coxeter number of X. Then u − r is conjugate to a norm 2 vector of II_{1,25} in D of height t − h + 1 (or t − h if the entry under “Dim” is 24E) whose letter is the letter corresponding to X in the column headed “norm 2”. For example let u be the vector of height 6 and root system a_{2}^{2}a_{1}^{10}. Then the norm 2 vectors corresponding to roots from the components a_{2} or a_{1} have heights 6 − 3 + 1 and 6 − 2 + 1 and letters a and b, so they are the vectors 4a and 5b of table 1.

If dim(L_{1}) ≤ 24 then the column “neighbors” gives the two even neighbors of L_{1} + I^{24−dim(L_{1})}. If dim(L_{1}) ≤ 23 then both neighbors are isomorphic so only one is listed, and if L_{1} is a Niemeier lattice then the neighbor is preceded by 2 (to indicate that the corresponding norm 0 vector is twice a primitive vector). If the two neighbors are isomorphic then there is an automorphism of L exchanging them.

Tables 1 and 2 were originally calculated by hand. Most of the lattices were found several times, once for each orbit of roots, and this gave a large number of checks for most entries. I later ran a computer version of the algorithm of this paper, which turned up about 20 minor errors (mostly errors in column 5, and a few misprints in the group order and root systems which were due to copying errors). I also checked the Minkowski-Siegel mass formula. Any errors remaining are probably either copying errors (the tables are based on computer output, but have had some hand editing to turn them into nice looking TEX) or an error where a lattice should be split into 2 lattices each with twice the automorphism group. The second possibility cannot be detected by mass formulas but I think it unlikely that it occurs in these tables. (It becomes an irritating problem when classifying the 26-dimensional even lattices of determinant 3.)

Height Dim  Roots  Group  norm 2 Neighbors
1 24E  None 8315553613086720000  2Λ
2 23  a_{1}^{2}  84610842624000  a  Λ
2 24  None  1002795171840  Λ  A_{1}^{24}
3 25  a_{1}^{2}  88704000  a
4 24  a_{1}^{8}  20643840  a  A_{1}^{24}  A_{1}^{24}

4 25  a_{2}^{2}  26127360  a
4 25  a_{1}^{6}  138240  a
5 24  a_{1}^{12}  190080  a  A_{1}^{24}  A_{2}^{12}
5 25  a_{2}a_{1}^{7}  5040  aa
5 25  a_{1}^{10}  1920  a
6 23  a_{1}^{16}a_{1}^{2}  645120  ca  A_{1}^{24}
6 24  a_{2}^{2}a_{1}^{10}  5760  ab  A_{2}^{12}  A_{2}^{12}
6 24  a_{1}^{16}  43008  c  A_{1}^{24}  A_{3}^{8}
6 25  a_{3}a_{1}^{8}  21504  ac
6 25  a_{2}^{2}a_{1}^{8}  128  ab
6 25  a_{2}a_{1}^{10}a_{1}  120  abc
6 25  a_{1}^{8}a_{1}^{6}  1152  bc
7 24  a_{2}^{4}a_{1}^{8}  384  bb  A_{2}^{12}  A_{3}^{8}
7 24E  a_{1}^{24}  244823040  a  2A_{1}^{24}
7 25  a_{2}^{5}a_{1}^{3}  720  bb
7 25  a_{3}a_{2}a_{1}^{9}  72  acb
7 25  a_{2}^{4}a_{1}^{4}a_{1}^{2}  24  bba
7 25  a_{3}a_{1}^{12}  1440  ab
7 25  a_{2}^{3}a_{1}^{6}a_{1}^{3}  12  bbb
7 25  a_{2}^{2}a_{1}^{12}  144  cb
8 22  a_{3}a_{1}^{22}  887040  ae  A_{1}^{24}
8 23  a_{2}^{6}a_{1}^{6}a_{1}^{2}  1440  bda  A_{2}^{12}
8 24  a_{3}^{2}a_{1}^{12}  768  cc  A_{3}^{8}  A_{3}^{8}
8 24  a_{3}a_{2}^{4}a_{1}^{6}  96  bbb  A_{3}^{8}  A_{3}^{8}
8 24  a_{2}^{8}  672  a  A_{3}^{8}  A_{3}^{8}
8 24  a_{2}^{6}a_{1}^{6}  240  bd  A_{2}^{12}  A_{4}^{6}
8 24  a_{1}^{24}  138240  e  A_{1}^{24}  D_{4}^{6}
8 25  a_{4}a_{1}^{12}  1440  ad
8 25  a_{3}a_{2}^{4}a_{1}^{4}  16  bbb
8 25  a_{3}^{2}a_{1}^{8}a_{1}^{2}  64  cbc
8 25  a_{3}a_{2}^{3}a_{1}^{6}a_{1}  12  bbbc
8 25  a_{3}a_{2}^{3}a_{1}^{3}a_{1}^{3}a_{1}  6  bbbbd
8 25  a_{2}^{4}a_{2}^{2}a_{1}^{4}  8  bab
8 25  a_{3}a_{2}^{2}a_{1}^{4}a_{1}^{4}a_{1}^{2}  16  bbcbd
8 25  a_{2}^{4}a_{2}a_{1}^{4}a_{1}^{2}a_{1}  8  bbbdb
8 25  a_{2}^{4}a_{1}^{8}a_{1}^{2}  48  bdc
8 25  a_{3}a_{1}^{15}a_{1}  720  cce
9 24  a_{3}^{2}a_{2}^{4}a_{1}^{4}  16  bbb  A_{3}^{8}  A_{4}^{6}
9 24  a_{2}^{8}a_{1}^{4}  384  dc  A_{2}^{12} A_{5}^{4}D_{4}

9 25  a_{4}a_{2}^{3}a_{1}^{6}a_{1}  12  bdbb
9 25  a_{3}^{2}a_{2}^{4}a_{1}^{2}  16  bba
9 25  a_{3}^{2}a_{2}^{2}a_{2}a_{1}^{4}a_{1}  4  bbcba
9 25  a_{3}a_{3}a_{2}^{2}a_{2}a_{1}^{2}a_{1}^{2}a_{1}  2 bbbbbbb
9 25  a_{3}a_{2}^{6}a_{1}^{2}  6  abb
9 25  a_{3}^{2}a_{2}^{2}a_{1}^{4}a_{1}^{4}  8  bcbb
9 25  a_{3}a_{2}^{4}a_{2}a_{1}^{4}a_{1}  8  bbbbc
9 25  a_{3}a_{2}^{2}a_{2}^{2}a_{2}a_{1}^{2}a_{1}^{2}a_{1}  2 bbbdbbb
10 22  a_{3}a_{2}^{10}  2880  ac  A_{2}^{12}
10 23  a_{3}^{4}a_{1}^{8}a_{1}^{2}  384  cfa  A_{3}^{8}
10 23  a_{3}^{3}a_{2}^{4}a_{1}^{2}a_{1}^{2}  48  bbea  A_{3}^{8}
10 24  a_{3}^{4}a_{2}^{2}a_{1}^{2}  32  bab  A_{4}^{6}  A_{4}^{6}
10 24  a_{4}a_{3}a_{2}^{4}a_{1}^{4}  16  bdbc  A_{4}^{6}  A_{4}^{6}
10 24  a_{3}^{2}a_{3}a_{2}^{4}a_{1}^{2}  16  bbbe  A_{3}^{8} A_{5}^{4}D_{4}
10 24  a_{3}^{4}a_{1}^{4}a_{1}^{4}  48  cdf  A_{3}^{8} A_{5}^{4}D_{4}
10 24  a_{3}^{4}a_{1}^{8}  384  cd  A_{3}^{8}  D_{4}^{6}
10 24E  a_{2}^{12}  190080  a  2A_{2}^{12}
10 25  a_{3}^{5}  1920  c
10 25  d_{4}a_{2}^{4}a_{1}^{6}  144  bcd
10 25  d_{4}a_{3}a_{1}^{12}  576  ced
10 25  a_{5}a_{1}^{15}  720  cf
10 25  a_{4}a_{3}a_{2}^{4}a_{1}^{2}  8  bdbb
10 25  a_{3}^{3}a_{3}a_{2}a_{1}^{3}  6  bcbb
10 25  a_{4}a_{3}a_{2}^{2}a_{2}a_{1}^{2}a_{1}^{2}a_{1}  2 bdbbbce
10 25  a_{3}^{3}a_{3}a_{1}^{4}a_{1}^{2}  24  bcce
10 25  a_{3}^{2}a_{3}^{2}a_{1}^{4}a_{1}^{2}  16  cbbd
10 25  a_{4}a_{2}^{6}a_{1}^{2}  6  abe
10 25  a_{4}a_{3}a_{2}^{2}a_{1}^{4}a_{1}^{2}a_{1}^{2}  4  bdbccf
10 25  a_{3}^{2}a_{3}a_{2}^{2}a_{2}a_{1}^{2}a_{1}  2 bbbbbc
10 25  a_{3}^{2}a_{3}a_{2}^{2}a_{2}a_{1}a_{1}a_{1}  2 bbbadbc
10 25  a_{4}a_{2}^{5}a_{1}^{5}  10  bbc
10 25  a_{3}^{2}a_{2}^{6}  48  da
10 25  a_{3}^{2}a_{2}^{4}a_{2}^{2}  8  bba
10 25  a_{3}^{3}a_{2}^{2}a_{1}^{3}a_{1}^{2}a_{1}  12  bbdcf
10 25 a_{3}a_{3}a_{3}a_{2}^{2}a_{1}^{2}a_{1}a_{1}a_{1}a_{1}  2 cbbbcebfd
10 25  a_{3}a_{3}a_{2}^{2}a_{2}^{2}a_{2}a_{1}^{2}a_{1}  2 bbbbbee
10 25  a_{3}^{2}a_{2}^{4}a_{1}^{4}a_{1}^{2}  8  dbcf
10 25  a_{3}^{3}a_{1}^{12}  48  cf
10 25  a_{3}a_{2}^{6}a_{2}a_{1}^{3}  12  dbce
11 24  a_{4}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}  4  bbbbb  A_{4}^{6} A_{5}^{4}D_{4}
11 24  a_{4}^{2}a_{2}^{4}a_{1}^{4}  16  dca  A_{4}^{6} A_{5}^{4}D_{4}
11 24  a_{3}^{6}  240  a  A_{4}^{6}  D_{4}^{6}

11 24  a_{3}^{4}a_{2}^{4}  24  be  A_{3}^{8}  A_{6}^{4}
11 25  a_{5}a_{2}^{4}a_{2}a_{1}^{4}  8  befb
11 25  d_{4}a_{3}a_{2}^{4}a_{1}^{4}  8  bcda
11 25  a_{4}a_{3}^{2}a_{3}a_{2}a_{1}^{2}a_{1}  2  bbbcba
11 25  a_{4}^{2}a_{2}^{2}a_{2}a_{1}^{4}a_{1}  4  dcbbb
11 25  a_{4}a_{3}^{2}a_{2}^{4}  4  bbb
11 25  a_{4}a_{3}^{3}a_{1}^{6}  6  cbb
11 25  a_{4}a_{3}^{2}a_{2}^{2}a_{2}a_{1}^{2}a_{1}  2  bbbfab
11 25 a_{4}a_{3}a_{3}a_{2}a_{2}a_{2}a_{1}a_{1}a_{1}  1 bbbbccbba
11 25 a_{4}a_{3}a_{3}a_{2}a_{2}a_{2}a_{1}a_{1}a_{1}  1 bbbbecbbb
11 25  a_{3}^{4}a_{3}a_{1}^{4}  8  bab
11 25  a_{3}^{2}a_{3}^{2}a_{2}^{2}a_{2}a_{1}  2  bbbbb
11 25  a_{3}^{2}a_{3}a_{3}a_{2}^{2}a_{2}a_{1}  2 bbabda
11 25  a_{4}a_{3}a_{2}^{2}a_{2}^{2}a_{2}a_{1}^{2}a_{1}  2 bbeccba
11 25  a_{3}^{2}a_{3}^{2}a_{2}^{2}a_{1}^{4}  4  bbdb
12 22  a_{3}^{6}a_{3}a_{1}^{2}  96  dag  A_{3}^{8}
12 23  a_{4}^{2}a_{3}^{2}a_{2}^{2}a_{1}^{2}a_{1}^{2}  8  bcbha  A_{4}^{6}
12 23  a_{4}a_{3}^{5}a_{1}^{2}  40  aba  A_{4}^{6}
12 24  d_{4}a_{4}a_{2}^{6}  24  dca  A_{5}^{4}D_{4} A_{5}^{4}D_{4}
12 24  d_{4}a_{3}^{4}a_{1}^{4}  32  cde  A_{5}^{4}D_{4} A_{5}^{4}D_{4}
12 24  a_{5}a_{3}^{3}a_{1}^{6}a_{1}  24  cfec  A_{5}^{4}D_{4} A_{5}^{4}D_{4}
12 24  a_{4}^{2}a_{3}^{2}a_{3}a_{1}^{2}  8  bbbd  A_{5}^{4}D_{4} A_{5}^{4}D_{4}
12 24  a_{5}a_{3}^{2}a_{2}^{4}a_{1}  16  bebf  A_{5}^{4}D_{4} A_{5}^{4}D_{4}
12 24  d_{4}a_{3}^{4}a_{1}^{4}  48  cdc  D_{4}^{6} A_{5}^{4}D_{4}
12 24  d_{4}^{2}a_{1}^{16}  1152  eb  D_{4}^{6}  D_{4}^{6}
12 24  a_{4}^{2}a_{3}^{2}a_{2}^{2}a_{1}^{2}  4  bcbh  A_{4}^{6}  A_{6}^{4}
12 24  a_{3}^{4}a_{3}^{2}a_{1}^{4}  32  fdg  A_{3}^{8} A_{7}^{2}D_{5}^{2}
12 25  a_{5}a_{3}^{2}a_{3}a_{1}^{4}a_{1}  8  cefdc
12 25  a_{5}a_{3}^{2}a_{2}^{2}a_{2}a_{1}a_{1}  2 bebbdh
12 25  d_{4}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}  4  bddac
12 25  d_{4}a_{4}a_{2}^{4}a_{1}^{4}  8  dcae
12 25  a_{5}a_{3}^{2}a_{2}^{2}a_{1}^{2}a_{1}^{2}a_{1}  4  bfbdhf
12 25  a_{5}a_{3}a_{3}a_{2}^{2}a_{1}^{2}a_{1}a_{1}a_{1}  2 bfebhdee
12 25  a_{4}^{2}a_{3}a_{3}a_{2}a_{1}^{2}a_{1}  2  bbcade
12 25  a_{4}a_{4}a_{3}a_{3}a_{2}a_{1}a_{1}a_{1}  1 bbccbddd
12 25  a_{4}^{2}a_{3}a_{2}^{4}  4  bbb
12 25  a_{4}^{2}a_{3}^{2}a_{1}^{4}a_{1}^{2}  4  bche
12 25  d_{4}a_{3}^{2}a_{2}^{4}a_{1}a_{1}  8  bdaeg
12 25  d_{4}a_{3}^{3}a_{1}^{6}a_{1}a_{1}  12  cdegb
12 25  a_{4}a_{3}^{2}a_{3}^{2}a_{2}a_{1}  2  abcbc
12 25  a_{4}^{2}a_{3}a_{2}^{2}a_{2}a_{1}a_{1}a_{1}  2 bcbbfdh
12 25 a_{4}a_{4}a_{3}a_{2}a_{2}a_{2}a_{1}a_{1}a_{1}  1 bbcabbhhd
12 25  a_{4}a_{3}^{4}a_{1}^{4}  8  ach

12 25  a_{4}a_{3}^{2}a_{3}a_{3}a_{1}^{2}a_{1}^{2}  2  bbcfde
12 25  a_{4}a_{3}^{3}a_{2}^{3}a_{1}  6  bbag
12 25  a_{4}a_{3}a_{3}a_{3}a_{2}a_{2}a_{2}a_{1}  1 bbbebbah
12 25  a_{3}^{4}a_{3}a_{3}a_{1}^{2}  8  bfdc
12 25  a_{4}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}a_{1}a_{1}  2 bccbhge
12 25  a_{4}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}a_{1}^{2}  2  bcfbhh
12 25  a_{3}^{2}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}  4  edbbh
13 24  a_{4}^{4}a_{1}^{4}  24  cc  A_{5}^{4}D_{4}  A_{6}^{4}
13 24  a_{5}a_{4}a_{3}a_{3}a_{2}a_{2}a_{1}  2 bebbddd  A_{5}^{4}D_{4}  A_{6}^{4}
13 24  a_{4}^{2}a_{3}^{4}  16  bb  A_{4}^{6} A_{7}^{2}D_{5}^{2}
13 24  a_{4}^{2}a_{4}a_{3}a_{2}^{2}a_{1}^{2}  4  ccahb  A_{4}^{6} A_{7}^{2}D_{5}^{2}
13 24E  a_{3}^{8}  2688  a  2A_{3}^{8}
13 25  a_{6}a_{2}^{6}a_{1}^{3}  12  dhd
13 25  a_{4}^{4}a_{1}^{2}  24  ca
13 25  a_{5}a_{4}a_{3}^{2}a_{2}a_{1}^{2}  2  bfbdc
13 25  d_{4}a_{4}a_{3}^{3}a_{1}^{2}  6  bdac
13 25  d_{4}^{2}a_{2}^{6}  72  cc
13 25  a_{5}a_{4}a_{3}a_{2}a_{2}a_{2}a_{1}a_{1}  1 bebhdeda
13 25  a_{5}a_{4}a_{3}a_{2}a_{2}a_{2}a_{1}a_{1}  1 bebhdddd
13 25  d_{4}a_{4}a_{3}a_{3}a_{2}a_{2}a_{1}a_{1}  1 bdaaeccb
13 25  a_{4}^{2}a_{4}a_{3}a_{2}^{2}  2  bcbd
13 25  a_{4}a_{4}a_{4}a_{3}a_{2}a_{1}a_{1}a_{1}  1 bccbdbcd
13 25  a_{5}a_{3}^{3}a_{2}^{3}  6  bbd
13 25  a_{5}a_{3}^{2}a_{3}a_{2}^{2}a_{2}  2  abbhc
13 25  a_{5}a_{4}a_{2}^{2}a_{2}^{2}a_{2}a_{1}^{2}  2  behdfd
13 25  a_{5}a_{3}^{2}a_{3}a_{2}^{2}a_{1}^{2}a_{1}  2 bbbedd
13 25  a_{4}a_{4}a_{3}a_{3}a_{3}a_{2}a_{1}  1 bcbbbdc
13 25  a_{4}a_{4}a_{3}a_{3}a_{3}a_{2}a_{1}  1 bbbabdb
13 25  a_{4}^{3}a_{2}^{3}a_{1}^{3}  6  cdb
13 25  d_{4}a_{3}^{3}a_{2}^{3}a_{2}  6  bacg
13 25  a_{4}^{2}a_{3}a_{3}a_{2}^{2}a_{2}a_{1}  2  ebaddd
13 25  a_{5}a_{2}^{9}  72  cf
14 21  d_{4}a_{3}^{7}  336  ag  A_{3}^{8}
14 22  a_{4}^{4}a_{3}a_{2}^{2}  16  aab  A_{4}^{6}
14 23  a_{5}a_{4}^{2}a_{3}a_{3}a_{1}^{2}a_{1}  4  bbddaf  A_{5}^{4}D_{4}
14 23  d_{4}a_{4}^{3}a_{2}^{2}a_{1}^{2}  12  caca  A_{5}^{4}D_{4}
14 23  a_{5}d_{4}a_{3}^{3}a_{1}^{3}a_{1}^{2}  12  dfega  A_{5}^{4}D_{4}
14 23  a_{5}^{2}a_{3}a_{2}^{4}a_{1}^{2}  16  efda  A_{5}^{4}D_{4}
14 23  d_{4}^{2}a_{3}^{4}a_{1}^{2}  96  dcd  D_{4}^{6}
14 24  a_{5}a_{4}^{3}a_{1}^{3}  12  cbe  A_{6}^{4}  A_{6}^{4}
14 24  a_{5}^{2}a_{3}^{2}a_{2}^{2}  8  eda  A_{6}^{4}  A_{6}^{4}
14 24  a_{6}a_{3}^{3}a_{2}^{3}  12  bhd  A_{6}^{4}  A_{6}^{4}

14 24  d_{4}a_{4}^{2}a_{4}a_{2}^{2}  4  caac  A_{5}^{4}D_{4} A_{7}^{2}D_{5}^{2}
14 24  a_{5}d_{4}a_{3}^{2}a_{3}a_{1}^{2}a_{1}  4  dfecbg  A_{5}^{4}D_{4} A_{7}^{2}D_{5}^{2}
14 24  a_{5}^{2}a_{3}^{2}a_{1}^{2}a_{1}^{2}a_{1}^{2}  8  febcg  A_{5}^{4}D_{4} A_{7}^{2}D_{5}^{2}
14 24  a_{5}a_{4}^{2}a_{3}a_{3}a_{1}  4  bbddf  A_{5}^{4}D_{4} A_{7}^{2}D_{5}^{2}
14 24  d_{4}^{2}a_{3}^{4}  32  dc  D_{4}^{6} A_{7}^{2}D_{5}^{2}
14 24  a_{4}^{3}a_{3}^{3}  12  bh  A_{4}^{6}  A_{8}^{3}
14 24  a_{3}^{8}  384  g  A_{3}^{8}  D_{6}^{4}
14 25  d_{5}a_{3}^{2}a_{2}^{4}a_{1}^{2}  8  bgbb
14 25  d_{5}a_{3}^{3}a_{1}^{8}  48  cgc
14 25  a_{6}a_{3}^{2}a_{3}a_{2}^{2}a_{1}  2  bhhcf
14 25  a_{5}^{2}a_{3}a_{3}a_{1}^{4}  8  efee
14 25  a_{6}a_{3}^{2}a_{3}a_{2}a_{1}^{2}a_{1}^{2}  2  bhhdeg
14 25  d_{4}a_{4}^{3}a_{1}^{3}a_{1}  6  cabc
14 25  a_{5}d_{4}a_{3}a_{3}a_{2}^{2}a_{1}  2  deeccb
14 25  a_{5}a_{5}a_{3}a_{2}^{2}a_{1}^{2}a_{1}a_{1}  2  efddefg
14 25  a_{5}a_{4}^{2}a_{3}a_{2}a_{1}^{2}  2  bbfdf
14 25  a_{5}a_{4}a_{4}a_{3}a_{2}a_{1}a_{1}  1 cbbdabe
14 25  d_{4}a_{4}^{2}a_{3}^{2}a_{1}^{2}  4  baeb
14 25  d_{4}^{2}a_{3}^{2}a_{3}a_{1}^{4}  8  dcbb
14 25  a_{5}a_{4}^{2}a_{3}a_{1}^{2}a_{1}^{2}a_{1}  2  cbhcee
14 25  a_{5}d_{4}a_{3}^{2}a_{1}^{4}a_{1}^{2}a_{1}  4  dfegcg
14 25  a_{5}a_{4}a_{3}a_{3}a_{3}a_{2}  1 bbhddc
14 25  a_{5}a_{4}a_{3}a_{3}a_{3}a_{2}  1  bbeddc
14 25  a_{5}a_{4}^{2}a_{2}^{2}a_{2}a_{1}^{2}  2  cbddf
14 25  a_{4}^{4}a_{2}^{2}  8  ba
14 25  d_{4}a_{4}a_{4}a_{3}a_{2}a_{2}a_{1}a_{1}  1 caaecbbg
14 25  a_{5}a_{4}a_{3}a_{3}a_{3}a_{1}a_{1}a_{1}  1 bbhedbfg
14 25  a_{5}a_{3}^{2}a_{3}^{2}a_{3}a_{1}  4  dddcg
14 25  a_{5}a_{3}^{2}a_{3}a_{3}a_{3}a_{1}  2  bheddf
14 25  a_{5}a_{4}a_{3}^{2}a_{2}^{2}a_{1}^{2}a_{1}  2  cbhdge
14 25  a_{4}^{2}a_{4}a_{3}^{2}a_{2}a_{1}  2  bbdbf
14 25  a_{4}a_{4}a_{4}a_{3}a_{3}a_{2}a_{1}  1 abbdhcf
14 25  a_{4}^{2}a_{4}a_{3}a_{2}^{2}a_{2}a_{1}  2  bahbdf
14 25  d_{4}a_{3}^{4}a_{3}a_{1}^{4}  8  fegg
15 24  a_{5}^{2}a_{4}^{2}a_{1}^{2}  4  bdc  A_{6}^{4} A_{7}^{2}D_{5}^{2}
15 24  a_{6}a_{4}a_{4}a_{3}a_{2}a_{1}a_{1}  2 chhceaa  A_{6}^{4} A_{7}^{2}D_{5}^{2}
15 24  a_{5}^{2}a_{4}a_{3}a_{2}^{2}  4  bfdf  A_{5}^{4}D_{4}  A_{8}^{3}
15 24  a_{4}^{4}a_{3}^{2}  16  hb  A_{4}^{6} A_{9}^{2}D_{6}
15 25  d_{5}a_{3}^{5}  20  ab
15 25  d_{5}a_{4}a_{3}^{2}a_{2}^{2}a_{1}^{2}  2  bgbba
15 25  a_{6}a_{4}^{2}a_{3}a_{1}^{2}a_{1}  2  chdcb
15 25  a_{6}a_{4}a_{4}a_{2}a_{2}a_{1}a_{1}a_{1}  1 chhefacc
15 25  a_{5}d_{4}a_{4}^{2}a_{1}^{2}a_{1}  2  abeab

15 25  a_{5}a_{5}a_{4}a_{3}a_{2}a_{1}  1  bbddea
15 25  a_{6}a_{4}a_{3}a_{3}a_{2}a_{2}a_{1}  1 bhdcfga
15 25  a_{6}a_{4}a_{3}^{2}a_{2}^{2}a_{1}  2  bhdfc
15 25  d_{4}^{2}a_{4}^{2}a_{2}^{2}  4  acb
15 25  a_{5}^{2}a_{4}a_{3}a_{1}^{2}a_{1}^{2}  2  bedcc
15 25  a_{5}d_{4}a_{4}a_{3}a_{2}a_{2}a_{1}  1 abecbba
15 25  a_{5}^{2}a_{3}^{2}a_{3}a_{1}^{2}  2  bdac
15 25  a_{5}a_{4}^{2}a_{4}a_{2}a_{1}^{2}  2  adfec
15 25  a_{5}a_{4}^{2}a_{4}a_{2}a_{1}^{2}  2  addca
15 25  a_{5}a_{4}a_{4}a_{4}a_{2}a_{1}a_{1}  1 bhddeca
15 25  a_{4}^{5}  5  d
15 25  a_{5}a_{4}a_{4}a_{3}a_{3}a_{2}  1  bhdcab
15 25  d_{4}a_{4}^{2}a_{3}^{2}a_{3}  2  bccb
16 21  d_{4}a_{4}^{5}  40  ab  A_{4}^{6}
16 22  a_{5}^{2}d_{4}a_{3}^{2}a_{3}  8  ceba  A_{5}^{4}D_{4}
16 22  a_{5}^{3}a_{3}a_{3}a_{1}^{3}  12  ecad  A_{5}^{4}D_{4}
16 22  d_{4}^{4}a_{3}a_{1}^{6}  144  bdc  D_{4}^{6}
16 23  a_{6}a_{5}a_{4}a_{3}a_{2}a_{1}^{2}a_{1}  2 bhdecah  A_{6}^{4}
16 23  a_{5}^{3}a_{4}a_{1}^{2}a_{1}  6  daag  A_{6}^{4}
16 24  d_{5}d_{4}a_{3}^{4}  16  dgb  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  d_{5}a_{5}a_{3}^{2}a_{3}a_{1}^{2}a_{1}  4  fgbceb  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  a_{5}^{2}d_{4}^{2}a_{1}^{2}  16  cef  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  a_{7}a_{3}^{4}a_{1}^{4}  16  fge  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  d_{5}a_{4}a_{4}^{2}a_{2}^{2}  4  cbba  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  a_{6}d_{4}a_{4}^{2}a_{2}  4  ahcb  A_{7}^{2}D_{5}^{2} A_{7}^{2}D_{5}^{2}
16 24  a_{6}a_{5}a_{4}a_{3}a_{2}a_{1}  2  bhdech  A_{6}^{4}  A_{8}^{3}
16 24  a_{5}^{2}d_{4}a_{3}^{2}a_{1}^{2}  4  eegd  A_{5}^{4}D_{4} A_{9}^{2}D_{6}
16 24  a_{5}a_{5}a_{4}^{2}a_{3}  4  dddf  A_{5}^{4}D_{4} A_{9}^{2}D_{6}
16 24  a_{5}^{2}d_{4}a_{3}^{2}a_{1}^{2}  8  eebd  A_{5}^{4}D_{4}  D_{6}^{4}
16 24  d_{4}^{4}a_{1}^{8}  48  bc  D_{4}^{6}  D_{6}^{4}
16 24E  a_{4}^{6}  240  a  2A_{4}^{6}
16 25  a_{7}a_{3}^{4}a_{1}^{2}  8  fff
16 25  a_{7}a_{3}a_{3}a_{3}a_{2}^{2}a_{1}^{2}  2  efggch
16 25  a_{5}d_{4}^{3}a_{1}^{3}  36  bcb
16 25  d_{5}a_{4}^{2}a_{3}a_{3}a_{1}a_{1}  2 bbcbdb
16 25  d_{5}d_{4}a_{3}^{3}a_{1}^{3}a_{1}  6  dgbec
16 25  d_{5}a_{5}a_{3}a_{2}^{4}a_{1}  4  egcad
16 25  a_{6}a_{5}a_{4}a_{2}a_{2}a_{1}a_{1}  1 bhdcagh
16 25  d_{5}a_{4}a_{3}^{4}  4  bbb
16 25  d_{5}a_{4}^{2}a_{3}a_{2}^{2}a_{1}^{2}  2  cbcae
16 25  a_{6}d_{4}a_{4}a_{3}a_{2}a_{1}a_{1}  1 ahcgafe
16 25  a_{5}^{2}a_{5}a_{3}a_{2}  2  deeb
16 25  a_{6}a_{5}a_{3}a_{3}a_{2}a_{2}  1  bhfebc

16 25  a_{6}a_{5}a_{3}a_{3}a_{2}a_{1}a_{1}a_{1}  1 bhegchhh
16 25  a_{6}a_{4}^{2}a_{3}^{2}a_{1}  2  bcee
16 25  a_{5}a_{5}a_{5}a_{2}^{2}a_{1}^{2}a_{1}  2  defchd
16 25  a_{5}^{2}d_{4}a_{3}a_{2}^{2}  4  cfcb
16 25  a_{5}a_{5}d_{4}a_{3}a_{2}^{2}  2  ccdga
16 25  a_{6}d_{4}a_{3}^{2}a_{2}^{2}a_{2}  2  ahgab
16 25  a_{5}a_{5}a_{4}a_{4}a_{2}a_{1}  1  ddddcg
16 25  a_{5}a_{5}a_{4}a_{4}a_{2}a_{1}  1 ddadad
16 25  a_{6}a_{4}a_{4}a_{3}a_{2}a_{2}a_{1}  1 bddecah
16 25  a_{5}d_{4}a_{4}^{2}a_{3}a_{1}  2  edccb
16 25  a_{5}d_{4}^{2}a_{3}^{2}a_{1}a_{1}a_{1}  2  cebecf
16 25  a_{5}^{2}a_{4}a_{3}^{2}a_{1}^{2}  2  haeh
16 25  a_{5}^{2}a_{4}a_{3}a_{3}a_{1}^{2}  2  ddebd
16 25  a_{5}a_{5}a_{4}a_{3}a_{3}a_{1}a_{1}  1 eddefdg
16 25  a_{5}^{2}a_{3}^{4}  8  cf
16 25  a_{5}^{2}a_{4}a_{3}a_{2}^{2}a_{1}^{2}  2  hdcch
16 25  d_{4}a_{4}^{2}a_{4}a_{3}^{2}  2  hcbb
17 24  a_{7}a_{4}^{2}a_{3}^{2}  4  bfc  A_{7}^{2}D_{5}^{2}  A_{8}^{3}
17 24  a_{6}^{2}a_{4}a_{3}a_{1}^{2}  4  hebb  A_{7}^{2}D_{5}^{2}  A_{8}^{3}
17 24  a_{6}a_{5}a_{5}a_{3}a_{2}  2  dddcg  A_{6}^{4} A_{9}^{2}D_{6}
17 24  a_{6}^{2}a_{3}^{2}a_{2}^{2}  4  hah  A_{6}^{4} A_{9}^{2}D_{6}
17 24  a_{5}^{4}  24  a  A_{6}^{4}  D_{6}^{4}
17 25  a_{7}a_{4}a_{4}a_{3}a_{2}a_{1}  1  bfgcgb
17 25  a_{6}^{2}a_{3}^{2}a_{2}a_{1}  2  hcfa
17 25  a_{7}a_{4}^{2}a_{2}^{2}a_{2}a_{1}  2  bfhea
17 25  d_{5}d_{4}a_{4}^{2}a_{2}^{2}  2  abbb
17 25  d_{5}a_{5}a_{4}a_{3}a_{3}a_{1}  1 bbbaab
17 25  d_{5}a_{5}a_{4}a_{3}a_{2}a_{2}a_{1}  1 bbbaedb
17 25  a_{6}a_{5}d_{4}a_{3}a_{2}a_{1}  1  ecdafb
17 25  a_{6}a_{5}a_{4}^{2}a_{1}^{2}  2  ddeb
17 25  a_{6}a_{5}a_{4}a_{3}a_{3}  1  dcgbc
17 25  a_{6}a_{5}a_{4}a_{3}a_{3}  1  dcfcc
17 25  a_{6}a_{5}a_{4}a_{3}a_{3}  1  dceac
17 25  a_{5}^{2}d_{4}a_{4}a_{3}  2  cdbb
17 25  a_{6}a_{4}a_{4}a_{4}a_{3}a_{1}  1  deffab
18 21  a_{5}^{3}d_{4}d_{4}a_{1}  12  bcab  A_{5}^{4}D_{4}
18 22  a_{6}^{2}a_{4}^{2}a_{3}  4  caa  A_{6}^{4}
18 23  a_{6}d_{5}a_{4}a_{4}a_{2}a_{1}^{2}  2  bhaaba  A_{7}^{2}D_{5}^{2}
18 23  a_{6}^{2}d_{4}a_{4}a_{1}^{2}  4  ceba  A_{7}^{2}D_{5}^{2}
18 23  d_{5}a_{5}^{2}d_{4}a_{1}^{2}a_{1}^{2}  4  ebcfa  A_{7}^{2}D_{5}^{2}
18 23  a_{7}a_{5}a_{4}^{2}a_{1}^{2}a_{1}  4  dfcag  A_{7}^{2}D_{5}^{2}
18 23  a_{7}d_{4}^{2}a_{3}^{2}a_{1}^{2}  8  cgfa  A_{7}^{2}D_{5}^{2}

18 23  d_{5}^{2}a_{3}^{4}a_{1}^{2}  16  gea  A_{7}^{2}D_{5}^{2}
18 24  a_{7}a_{5}^{2}a_{2}^{2}  8  ffa  A_{8}^{3}  A_{8}^{3}
18 24  a_{7}a_{5}a_{4}^{2}a_{1}  4  dfcg  A_{7}^{2}D_{5}^{2} A_{9}^{2}D_{6}
18 24  a_{7}a_{5}d_{4}a_{3}a_{1}a_{1}a_{1}  2  eggfdfc  A_{7}^{2}D_{5}^{2} A_{9}^{2}D_{6}
18 24  a_{6}d_{5}a_{4}a_{4}a_{2}  2  bhaab  A_{7}^{2}D_{5}^{2} A_{9}^{2}D_{6}
18 24  d_{5}^{2}a_{3}^{4}  16  gb  A_{7}^{2}D_{5}^{2}  D_{6}^{4}
18 24  d_{5}a_{5}^{2}d_{4}a_{1}^{2}  4  ebbc  A_{7}^{2}D_{5}^{2}  D_{6}^{4}
18 24  a_{5}^{2}a_{5}d_{4}a_{3}a_{1}  4  gbcdb  A_{5}^{4}D_{4}A_{11}D_{7}E_{6}
18 24  a_{5}^{4}a_{1}^{4}  48  cb  A_{5}^{4}D_{4}  E_{6}^{4}
18 25  d_{6}a_{3}^{4}a_{3}a_{1}^{2}  8  ddcd
18 25  a_{7}a_{5}^{2}a_{1}^{4}  8  fge
18 25  a_{7}a_{5}d_{4}a_{2}^{2}a_{1}  2  egfbc
18 25  a_{7}d_{4}^{2}a_{3}a_{1}^{4}  4  cgef
18 25  a_{7}a_{5}a_{4}a_{3}a_{2}  1  dfchb
18 25  a_{6}^{2}a_{5}a_{2}a_{1}a_{1}  2  deaed
18 25  a_{7}a_{5}a_{4}a_{3}a_{1}a_{1}a_{1}  1 dgchfeg
18 25  a_{6}d_{5}a_{4}a_{3}a_{2}a_{1}a_{1}  1 bhaebfd
18 25  d_{5}a_{5}a_{5}a_{4}a_{1}a_{1}  1  dbcacc
18 25  a_{7}a_{5}a_{3}^{2}a_{3}a_{1}  2  dggfd
18 25  a_{7}a_{5}a_{3}^{2}a_{3}a_{1}  2  dfhfg
18 25  a_{6}a_{6}a_{4}a_{3}a_{2}a_{1}  1  cdchbg
18 25  a_{6}a_{5}a_{5}a_{4}a_{1}  1  aeecc
18 25  d_{5}a_{5}d_{4}a_{3}^{2}a_{1}^{2}a_{1}  2  ebcefb
18 25  d_{5}d_{4}^{2}a_{3}^{2}a_{3}  4  cbbc
18 25  d_{5}a_{5}a_{4}^{2}a_{3}a_{1}  2  dbadf
18 25  d_{5}a_{5}a_{4}^{2}a_{3}a_{1}  2  dbabb
18 25  a_{6}a_{5}d_{4}a_{4}a_{2}a_{1}  1  cgeabf
18 25  a_{6}a_{5}a_{4}a_{4}a_{3}  1  cfacg
18 25  a_{5}^{2}d_{4}^{2}a_{3}a_{1}^{2}  2  bgff
18 25  a_{6}a_{4}^{2}a_{4}^{2}a_{1}  2  bcag
19 24  a_{8}a_{4}^{2}a_{3}^{2}  4  hhb  A_{8}^{3} A_{9}^{2}D_{6}
19 24  a_{7}a_{6}a_{5}a_{2}a_{1}  2  dfceb  A_{8}^{3} A_{9}^{2}D_{6}
19 24  a_{6}a_{6}a_{5}a_{4}a_{1}  2  eeaha  A_{6}^{4}A_{11}D_{7}E_{6}
19 24E  d_{4}^{6}  2160  d  2D_{4}^{6}
19 24E  a_{5}^{4}d_{4}  48  aa 2A_{5}^{4}D_{4}
19 25  d_{6}a_{4}^{2}a_{4}a_{2}^{2}  2  addc
19 25  a_{8}a_{4}a_{4}a_{3}a_{2}a_{1}  1  hghbfb
19 25  a_{8}a_{4}^{2}a_{2}^{2}a_{2}a_{1}  2  hhgeb
19 25  a_{7}a_{6}a_{4}a_{2}a_{2}a_{1}  1  dfheeb
19 25  d_{5}^{2}a_{4}a_{4}a_{2}^{2}  2  bbec
19 25  a_{6}d_{5}d_{4}a_{4}a_{2}  1  bcaec
19 25  a_{6}d_{5}a_{5}a_{3}a_{2}a_{1}  1  bdabca
19 25  a_{7}a_{5}^{2}a_{3}a_{1}^{2}  2  dcab

19 25  d_{5}a_{5}^{3}a_{1}  3  aaa
19 25  a_{7}a_{5}^{2}a_{2}^{2}a_{1}^{2}  2  dcdb
19 25  a_{7}a_{5}a_{4}a_{4}a_{2}  1  achgc
19 25  a_{7}d_{4}a_{4}a_{4}a_{3}  1  ccefb
19 25  a_{6}^{2}a_{5}a_{3}a_{2}  2  fbad
19 25  a_{6}a_{5}^{3}  6  cb
19 25  a_{6}a_{5}^{2}a_{5}  2  ecb
19 25  a_{6}a_{5}a_{5}d_{4}a_{2}  1  babcc
19 25  d_{5}a_{5}^{2}a_{4}a_{2}^{2}  2  dadc
20 20  d_{5}a_{5}^{4}  16  ad  A_{5}^{4}D_{4}
20 20  d_{5}d_{4}^{5}  120  dc  D_{4}^{6}
20 21  a_{6}^{3}d_{4}a_{2}  6  aaa  A_{6}^{4}
20 22  a_{7}d_{5}a_{5}a_{3}a_{3}a_{1}  2  bgecad  A_{7}^{2}D_{5}^{2}
20 22  d_{5}^{2}a_{5}^{2}a_{3}  8  bba  A_{7}^{2}D_{5}^{2}
20 22  a_{7}^{2}a_{3}^{2}a_{3}a_{1}^{2}  8  gdae  A_{7}^{2}D_{5}^{2}
20 23  a_{7}a_{6}^{2}a_{1}^{2}a_{1}^{2}  4  ecga  A_{8}^{3}
20 23  a_{7}^{2}a_{4}a_{3}a_{1}^{2}  4  faea  A_{8}^{3}
20 23  a_{8}a_{5}^{2}a_{2}^{2}a_{1}^{2}  4  dhba  A_{8}^{3}
20 24  a_{7}^{2}d_{4}a_{1}^{4}  8  gff  A_{9}^{2}D_{6} A_{9}^{2}D_{6}
20 24  d_{6}a_{5}^{2}a_{3}^{2}  8  edd  A_{9}^{2}D_{6} A_{9}^{2}D_{6}
20 24  a_{8}a_{5}^{2}a_{3}  4  dge  A_{9}^{2}D_{6} A_{9}^{2}D_{6}
20 24  d_{6}a_{5}^{2}a_{3}^{2}  4  edc  D_{6}^{4} A_{9}^{2}D_{6}
20 24  d_{6}d_{4}^{3}a_{1}^{6}  12  bcb  D_{6}^{4}  D_{6}^{4}
20 24  a_{7}d_{5}a_{5}a_{3}a_{1}a_{1}a_{1}  2  cgefecd  A_{7}^{2}D_{5}^{2}A_{11}D_{7}E_{6}
20 24  d_{5}d_{5}a_{5}^{2}a_{1}^{2}  4  bced  A_{7}^{2}D_{5}^{2}A_{11}D_{7}E_{6}
20 24  a_{7}d_{5}d_{4}a_{3}a_{3}  2  bgecf  A_{7}^{2}D_{5}^{2}A_{11}D_{7}E_{6}
20 24  a_{6}a_{6}d_{5}a_{4}  2  aaeb  A_{7}^{2}D_{5}^{2}A_{11}D_{7}E_{6}
20 24  d_{5}^{2}a_{5}^{2}a_{1}^{2}  8  cbc  A_{7}^{2}D_{5}^{2}  E_{6}^{4}
20 24  a_{6}^{2}a_{5}^{2}  4  ch  A_{6}^{4}  A_{12}^{2}
20 24  d_{4}^{6}  48  c  D_{4}^{6}  D_{8}^{3}
20 25  a_{8}a_{5}^{2}a_{1}^{2}a_{1}^{2}  2  dhfg
20 25  d_{6}a_{5}^{2}a_{3}a_{1}^{2}a_{1}a_{1}  2  eddfeb
20 25  d_{6}a_{5}d_{4}a_{3}^{2}a_{1}  2  ccdcd
20 25  a_{7}a_{7}a_{3}a_{2}^{2}a_{1}^{2}  2  fggbg
20 25  a_{8}d_{4}a_{4}a_{3}a_{3}  1  chbff
20 25  d_{5}^{2}a_{5}d_{4}a_{1}^{2}a_{1}  2  bbecb
20 25  a_{7}d_{5}a_{4}^{2}a_{1}a_{1}  2  cfbcf
20 25  a_{7}a_{6}a_{5}a_{3}  1  ecfe
20 25  a_{6}^{2}d_{5}a_{3}a_{1}^{2}  2  aedd
20 25  a_{7}d_{5}a_{4}a_{3}a_{3}  1  bfbfc
20 25  a_{7}a_{6}a_{5}a_{2}a_{1}a_{1}a_{1}  1  echbgfe
20 25  a_{7}a_{6}a_{4}a_{4}a_{1}  1  ecbad
20 25  a_{6}^{2}a_{6}a_{3}a_{1}  2  abee

20 25  a_{6}a_{6}a_{6}a_{3}a_{1}  1  caceg
20 25  a_{6}d_{5}a_{5}a_{4}a_{2}a_{1}  1  aeebad
20 25  a_{6}a_{6}a_{5}d_{4}a_{1}  1  abfhd
20 25  a_{7}a_{5}^{2}a_{4}a_{1}^{2}  2  ehag
20 25  d_{5}a_{5}a_{5}a_{4}^{2}  2  fbdb
21 24  a_{8}a_{6}a_{5}a_{2}a_{1}  2  ehbga  A_{8}^{3}A_{11}D_{7}E_{6}
21 24  a_{7}^{2}a_{4}^{2}  4  cg  A_{7}^{2}D_{5}^{2}  A_{12}^{2}
21 25  d_{6}a_{6}a_{4}a_{4}a_{2}  1  cdcdd
21 25  a_{8}a_{6}a_{4}a_{3}a_{1}  1  eggbb
21 25  a_{7}a_{6}d_{5}a_{2}a_{1}a_{1}  1  aecfba
21 25  a_{6}d_{5}d_{5}a_{4}a_{2}  1  baacc
21 25  a_{7}d_{5}a_{5}a_{4}a_{1}  1  acbdb
21 25  a_{7}a_{6}a_{5}a_{4}  1  cgbe
22 21  a_{7}d_{5}^{2}d_{4}a_{3}  4  beac  A_{7}^{2}D_{5}^{2}
22 22  a_{8}a_{6}^{2}a_{3}  4  bba  A_{8}^{3}
22 23  a_{8}a_{7}a_{5}a_{1}^{2}a_{1}  2  cgeac  A_{9}^{2}D_{6}
22 23  a_{8}a_{6}d_{5}a_{2}a_{1}^{2}  2  abhba  A_{9}^{2}D_{6}
22 23  a_{7}d_{6}a_{5}a_{3}a_{1}^{2}a_{1}  2  dgdfab  A_{9}^{2}D_{6}
22 23  a_{9}a_{5}a_{4}^{2}a_{1}^{2}  4  fgba  A_{9}^{2}D_{6}
22 23  d_{6}d_{5}a_{5}^{2}a_{1}^{2}  4  bdcd  D_{6}^{4}
22 23  d_{5}^{4}a_{1}^{2}  48  bd  D_{6}^{4}
22 24  a_{9}a_{5}d_{4}a_{3}a_{1}a_{1}  2  gfffeb  A_{9}^{2}D_{6}A_{11}D_{7}E_{6}
22 24  a_{7}d_{6}a_{5}a_{3}a_{1}  2  dgcfe  A_{9}^{2}D_{6}A_{11}D_{7}E_{6}
22 24  a_{8}a_{6}d_{5}a_{2}  2  abhb  A_{9}^{2}D_{6}A_{11}D_{7}E_{6}
22 24  d_{6}d_{5}a_{5}^{2}  4  bdc  D_{6}^{4}A_{11}D_{7}E_{6}
22 24  d_{5}^{4}  48  b  D_{6}^{4}  E_{6}^{4}
22 24  a_{8}a_{7}a_{4}a_{3}  2  chbg  A_{8}^{3}  A_{12}^{2}
22 24  a_{7}d_{5}^{2}a_{3}^{2}  4  eed  A_{7}^{2}D_{5}^{2}  D_{8}^{3}
22 24  a_{7}^{2}d_{4}^{2}  8  fd  A_{7}^{2}D_{5}^{2}  D_{8}^{3}
22 24E  a_{6}^{4}  24  a  2A_{6}^{4}
22 25  e_{6}d_{4}^{2}a_{3}^{3}  12  cbc
22 25  e_{6}a_{5}a_{4}^{2}a_{3}a_{1}  2  dbace
22 25  e_{6}a_{5}^{2}a_{2}^{4}  8  fba
22 25  d_{7}a_{3}^{6}  24  ge
22 25  a_{9}a_{5}d_{4}a_{2}^{2}  2  gfgb
22 25  a_{9}a_{5}a_{4}a_{3}a_{1}a_{1}  1  ffbgbc
22 25  a_{9}a_{5}a_{3}^{2}a_{3}  2  fggf
22 25  a_{8}a_{7}a_{4}a_{2}a_{1}  1  chbbc
22 25  d_{6}d_{5}a_{5}a_{3}a_{3}a_{1}  1  bdcdbe
22 25  a_{7}d_{6}a_{3}^{2}a_{3}a_{1}^{2}  2  dgfeb
22 25  a_{8}d_{5}a_{5}a_{3}a_{1}  1  agffe
22 25  a_{8}d_{5}a_{5}a_{2}a_{2}a_{1}  1  ahfbab

22 25  a_{8}a_{6}a_{5}a_{3}  1  bbgg
22 25  a_{8}a_{6}a_{5}a_{3}  1  cbef
22 25  d_{6}a_{5}^{2}d_{4}a_{3}  2  bcdb
22 25  d_{6}a_{5}^{3}a_{1}^{3}  6  cdb
22 25  a_{8}a_{5}^{2}d_{4}  2  bfe
22 25  a_{8}a_{6}a_{4}a_{4}a_{1}  1  cbbbc
22 25  a_{7}a_{7}a_{5}a_{3}a_{1}  1  ghefc
22 25  d_{5}^{3}a_{3}^{3}  6  ec
22 25  a_{7}d_{5}d_{4}^{2}a_{3}  2  bffc
23 24  a_{8}^{2}a_{3}^{2}  4  hb  A_{9}^{2}D_{6}  A_{12}^{2}
23 24  a_{9}a_{6}a_{5}a_{2}  2  cgbc  A_{9}^{2}D_{6}  A_{12}^{2}
23 24  a_{7}^{3}  12  a  A_{8}^{3}  D_{8}^{3}
23 25  d_{7}a_{4}^{4}  4  bd
23 25  a_{9}a_{6}a_{4}a_{3}  1  cfgb
23 25  a_{8}d_{5}^{2}a_{2}^{2}  2  ebe
23 25  d_{6}a_{6}a_{6}a_{4}  1  accf
23 25  d_{6}a_{6}^{2}a_{4}  2  bce
23 25  a_{8}a_{7}d_{4}a_{3}  1  fbbb
23 25  a_{7}^{2}d_{5}a_{3}  2  baa
23 25  a_{6}^{2}d_{5}^{2}  2  cb
24 20  a_{7}^{2}d_{5}d_{5}  4  cda  A_{7}^{2}D_{5}^{2}
24 21  a_{8}^{2}d_{4}a_{4}  4  baa  A_{8}^{3}
24 22  a_{9}a_{7}d_{4}a_{3}a_{1}  2  fffad  A_{9}^{2}D_{6}
24 22  a_{7}^{2}d_{6}a_{3}  4  cfa  A_{9}^{2}D_{6}
24 22  d_{6}^{2}d_{4}^{2}a_{3}a_{1}^{2}  4  cbdb  D_{6}^{4}
24 24  d_{7}a_{5}^{2}a_{5}a_{1}  4  cdea A_{11}D_{7}E_{6}A_{11}D_{7}E_{6}
24 24  a_{9}d_{5}^{2}a_{1}^{2}a_{1}  4  efec A_{11}D_{7}E_{6}A_{11}D_{7}E_{6}
24 24  a_{7}e_{6}d_{4}a_{3}^{2}  4  bgce A_{11}D_{7}E_{6}A_{11}D_{7}E_{6}
24 24  e_{6}a_{6}^{2}a_{4}  4  eaa A_{11}D_{7}E_{6}A_{11}D_{7}E_{6}
24 24  e_{6}d_{5}a_{5}^{2}a_{1}^{2}  4  cbca  E_{6}^{4}A_{11}D_{7}E_{6}
24 24  d_{6}^{2}d_{4}^{2}a_{1}^{4}  4  cbb  D_{6}^{4}  D_{8}^{3}
24 24  a_{7}^{2}d_{6}a_{1}^{2}  4  dfd  A_{9}^{2}D_{6}  D_{8}^{3}
24 24  a_{7}^{2}d_{5}d_{4}  4  fde  A_{7}^{2}D_{5}^{2} A_{15}D_{9}
24 25  d_{7}a_{5}^{2}d_{4}a_{1}a_{1}  2  bdeeb
24 25  a_{9}a_{7}a_{3}^{2}a_{1}  2  fgcd
24 25  d_{6}d_{5}^{2}a_{5}a_{1}  2  bcba
24 25  a_{7}d_{6}d_{5}a_{3}a_{1}a_{1}  1  cedeeb
24 25  a_{8}a_{7}a_{6}a_{1}  1  aebd
24 25  a_{7}d_{6}a_{5}d_{4}a_{1}  1  cfdfb
24 25  a_{7}^{2}a_{7}a_{1}^{2}  2  edd
24 25  a_{8}a_{7}d_{4}a_{4}  1  bfgb
24 25  a_{7}a_{7}d_{5}a_{4}  1  ccgb

24 25  a_{7}^{2}d_{5}a_{4}  2  cea
25 24  a_{10}a_{6}a_{5}a_{1}  2  hgbb A_{11}D_{7}E_{6}  A_{12}^{2}
25 24  a_{8}a_{7}^{2}  4  eb  A_{8}^{3} A_{15}D_{9}
25 24E  a_{7}^{2}d_{5}^{2}  8  aa 2A_{7}^{2}D_{5}^{2}
25 25  d_{7}a_{6}a_{6}a_{2}a_{2}  1  adedc
25 25  a_{10}a_{6}a_{4}a_{2}a_{1}  1  hgcea
25 25  e_{6}a_{6}d_{5}a_{4}a_{2}  1  acaea
25 25  e_{6}a_{6}^{2}d_{4}  2  bca
25 25  a_{7}e_{6}a_{5}a_{4}a_{1}  1  acaeb
25 25  a_{10}a_{5}a_{4}a_{4}  1  gbcb
25 25  a_{8}d_{6}a_{6}a_{2}  1  dbfc
25 25  a_{9}a_{6}d_{5}a_{2}a_{1}  1  bfbeb
25 25  a_{9}a_{6}^{2}a_{2}  2  agc
25 25  a_{9}d_{5}a_{5}a_{4}  1  bbbe
26 19  a_{7}^{2}d_{6}d_{5}  4  dae  A_{7}^{2}D_{5}^{2}
26 21  a_{9}d_{6}a_{5}d_{4}  2  cfea  A_{9}^{2}D_{6}
26 23  a_{9}d_{6}d_{5}a_{1}^{2}a_{1}  2  cffab A_{11}D_{7}E_{6}
26 23  a_{10}a_{6}d_{5}a_{1}^{2}  2  bbga A_{11}D_{7}E_{6}
26 23  a_{8}e_{6}a_{6}a_{2}a_{1}^{2}  2  ahaba A_{11}D_{7}E_{6}
26 23  d_{7}a_{7}d_{5}a_{3}a_{1}^{2}  2  edeea A_{11}D_{7}E_{6}
26 23  e_{6}d_{6}a_{5}^{2}a_{1}^{2}  4  dbea A_{11}D_{7}E_{6}
26 23  e_{6}d_{5}^{3}a_{1}^{2}  12  bce  E_{6}^{4}
26 24  a_{9}^{2}a_{2}^{2}  8  ga  A_{12}^{2}  A_{12}^{2}
26 24  d_{7}a_{7}d_{5}a_{3}  2  eddc A_{11}D_{7}E_{6}  D_{8}^{3}
26 24  a_{9}d_{6}a_{5}a_{3}  2  dfbd  A_{9}^{2}D_{6} A_{15}D_{9}
26 24  a_{9}a_{8}a_{5}  2  ebc  A_{9}^{2}D_{6} A_{15}D_{9}
26 24  a_{7}^{2}d_{5}^{2}  8  ce  A_{7}^{2}D_{5}^{2} D_{10}E_{7}^{2}
26 25  d_{7}d_{5}^{2}a_{3}a_{3}  2  bdba
26 25  a_{10}d_{5}a_{5}a_{2}a_{1}  1  bgbbb
26 25  a_{7}d_{6}^{2}a_{3}  2  bcc
26 25  a_{9}d_{6}d_{4}a_{3}a_{1}  1  cfbeb
26 25  a_{9}a_{7}a_{6}  1  egb
26 25  a_{9}a_{7}a_{6}  1  efb
26 25  a_{7}d_{6}d_{5}a_{5}a_{1}  1  dffeb
27 24  a_{8}^{2}a_{7}  4  ga  A_{8}^{3} A_{17}E_{7}
27 25  a_{8}d_{7}a_{4}a_{4}  1  dbde
27 25  a_{9}^{2}a_{3}a_{1}^{2}  2  baa
27 25  a_{10}a_{6}^{2}a_{1}  2  eca
28 20  a_{9}^{2}d_{5}a_{1}^{2}  4  fac  A_{9}^{2}D_{6}
28 20  d_{6}^{3}d_{5}a_{1}^{2}  6  bdb  D_{6}^{4}

28 22  a_{9}e_{6}d_{5}a_{3}a_{1}  2  cfead A_{11}D_{7}E_{6}
28 22  a_{9}d_{7}a_{5}a_{3}  2  dfca A_{11}D_{7}E_{6}
28 22  a_{11}d_{5}a_{5}a_{3}a_{1}  2  fbeae A_{11}D_{7}E_{6}
28 22  e_{6}^{2}a_{5}^{2}a_{3}  8  bae  E_{6}^{4}
28 23  a_{10}a_{9}a_{2}a_{1}^{2}a_{1}  2  bgaaf  A_{12}^{2}
28 23  a_{11}a_{7}a_{4}a_{1}^{2}  2  gcaa  A_{12}^{2}
28 24  d_{8}d_{4}^{4}  8  cb  D_{8}^{3}  D_{8}^{3}
28 24  a_{9}d_{7}a_{5}a_{1}a_{1}  2  efede A_{11}D_{7}E_{6} A_{15}D_{9}
28 24  a_{11}d_{5}d_{4}a_{3}  2  fbeb A_{11}D_{7}E_{6} A_{15}D_{9}
28 24  a_{9}a_{7}d_{6}a_{1}  2  fefc  A_{9}^{2}D_{6} A_{17}E_{7}
28 24  a_{9}a_{7}d_{6}a_{1}  2  fbfc  A_{9}^{2}D_{6} D_{10}E_{7}^{2}
28 24  d_{6}^{2}d_{6}d_{4}a_{1}^{2}  2  bbbb  D_{6}^{4} D_{10}E_{7}^{2}
28 24E  a_{8}^{3}  12  a  2A_{8}^{3}
28 25  d_{8}a_{5}^{2}a_{5}a_{1}  2  ddbe
28 25  a_{11}a_{7}a_{2}^{2}a_{1}^{2}  2  gbaf
28 25  a_{11}d_{5}a_{4}a_{3}  1  fcbb
28 25  e_{6}d_{6}d_{5}a_{5}a_{1}  1  cceab
28 25  d_{7}d_{6}a_{5}a_{5}  1  cdcb
28 25  a_{8}a_{7}e_{6}a_{1}a_{1}  1  aeedc
28 25  a_{9}e_{6}a_{4}^{2}a_{1}  2  cgbc
28 25  a_{10}a_{8}a_{4}a_{1}  1  bbae
28 25  a_{9}^{2}a_{4}a_{1}^{2}  2  gaf
28 25  a_{9}a_{8}d_{5}a_{1}  1  fbcd
29 24  a_{11}a_{8}a_{3}  2  bca  A_{12}^{2} A_{15}D_{9}
29 25  a_{10}d_{6}a_{6}  1  fbe
30 21  d_{7}a_{7}e_{6}d_{4}  2  cada A_{11}D_{7}E_{6}
30 22  a_{10}^{2}a_{3}  4  ba  A_{12}^{2}
30 23  d_{7}^{2}a_{7}a_{1}^{2}  4  dcd  D_{8}^{3}
30 23  d_{8}a_{7}^{2}a_{1}^{2}  4  ddd  D_{8}^{3}
30 24  d_{8}a_{7}^{2}  4  dd  D_{8}^{3} A_{15}D_{9}
30 24  a_{11}d_{6}a_{5}a_{1}  2  fbba A_{11}D_{7}E_{6} A_{17}E_{7}
30 24  a_{10}e_{6}a_{6}  2  agb A_{11}D_{7}E_{6} A_{17}E_{7}
30 24  d_{7}a_{7}e_{6}a_{3}  2  caed A_{11}D_{7}E_{6} D_{10}E_{7}^{2}
30 24  a_{9}e_{6}d_{6}a_{1}  2  efea A_{11}D_{7}E_{6} D_{10}E_{7}^{2}
30 24  e_{6}^{2}d_{5}^{2}  8  ca  E_{6}^{4} D_{10}E_{7}^{2}
30 25  d_{8}a_{7}d_{5}a_{3}  1  cbdd
30 25  a_{10}a_{9}a_{4}  1  bca
30 25  d_{7}a_{7}d_{5}^{2}  2  cae
31 24  a_{12}a_{7}a_{4}  2  gbf  A_{12}^{2} A_{17}E_{7}
31 24E  d_{6}^{4}  24  d  2D_{6}^{4}
31 24E  a_{9}^{2}d_{6}  4  aa 2A_{9}^{2}D_{6}

31 25  a_{12}a_{6}a_{5}  1  gba
31 25  a_{8}e_{6}^{2}a_{2}^{2}  2  aaa
31 25  a_{10}e_{6}d_{5}a_{2}  1  ebba
31 25  a_{11}a_{8}d_{4}  1  bea
31 25  a_{11}a_{7}d_{5}  1  bba
31 25  a_{8}a_{8}d_{7}  1  dcb
31 25  a_{8}^{2}d_{7}  2  ca
32 18  a_{9}^{2}d_{7}  4  da  A_{9}^{2}D_{6}
32 18  d_{7}d_{6}^{3}  6  db  D_{6}^{4}
32 20  a_{11}e_{6}d_{5}a_{3}  2  ebaa A_{11}D_{7}E_{6}
32 21  a_{12}a_{8}d_{4}  2  bba  A_{12}^{2}
32 22  d_{8}d_{6}^{2}a_{3}a_{1}^{2}  2  bbdb  D_{8}^{3}
32 24  d_{8}d_{6}^{2}a_{1}^{2}a_{1}^{2}  2  bbba  D_{8}^{3} D_{10}E_{7}^{2}
32 24  d_{6}^{4}  8  b  D_{6}^{4}  D_{12}^{2}
32 25  a_{9}d_{8}a_{5}a_{1}a_{1}  1  dfecb
32 25  a_{12}a_{7}d_{4}  1  bbf
32 25  a_{9}d_{7}d_{6}a_{1}  1  ceeb
34 19  a_{11}d_{7}d_{6}  2  cea A_{11}D_{7}E_{6}
34 19  e_{6}^{3}d_{6}  12  ae  E_{6}^{4}
34 23  a_{11}d_{8}a_{3}a_{1}^{2}  2  dbca  A_{15}D_{9}
34 23  a_{13}a_{8}a_{1}^{2}a_{1}  2  caac  A_{15}D_{9}
34 23  d_{9}a_{7}^{2}a_{1}^{2}  4  eea  A_{15}D_{9}
34 24  a_{13}d_{6}a_{3}a_{1}  2  bbcb  A_{15}D_{9} A_{17}E_{7}
34 24  d_{9}a_{7}^{2}  4  ed  A_{15}D_{9} D_{10}E_{7}^{2}
34 24  a_{11}d_{7}d_{5}  2  eee A_{11}D_{7}E_{6}  D_{12}^{2}
34 25  e_{7}a_{7}d_{5}a_{5}  1  cbca
34 25  e_{7}a_{7}^{2}a_{3}a_{1}  2  dcab
34 25  d_{9}a_{7}d_{5}a_{3}  1  ddeb
34 25  a_{13}a_{7}a_{3}a_{1}  1  cfcc
34 25  d_{7}^{2}e_{6}a_{3}  2  aca
34 25  d_{8}a_{7}e_{6}a_{3}  1  edda
34 25  a_{11}d_{6}^{2}  2  cb
35 24  a_{11}^{2}  4  a  A_{12}^{2}  D_{12}^{2}
35 25  a_{12}d_{7}a_{4}  1  ebc
36 20  d_{8}^{2}d_{5}d_{4}  2  bda  D_{8}^{3}
36 22  a_{13}d_{7}a_{3}a_{1}  2  ebab  A_{15}D_{9}
36 24  a_{9}e_{7}a_{7}  2  cfa D_{10}E_{7}^{2} A_{17}E_{7}
36 24  e_{7}d_{6}d_{6}d_{4}a_{1}  2  bbbaa D_{10}E_{7}^{2} D_{10}E_{7}^{2}
36 24  d_{8}^{2}d_{4}^{2}  2  bb  D_{8}^{3}  D_{12}^{2}
36 25  a_{12}a_{10}a_{1}  1  aab

37 24E  e_{6}^{4}  48  e  2E_{6}^{4}
37 24E  a_{11}d_{7}e_{6}  2  aaa 2A_{11}D_{7}E_{6}
37 25  a_{13}e_{6}a_{4}a_{1}  1  baba
38 17  a_{11}d_{8}e_{6}  2  dae A_{11}D_{7}E_{6}
38 21  a_{11}d_{9}d_{4}  2  dea  A_{15}D_{9}
38 23  d_{9}e_{6}^{2}a_{1}^{2}  4  add D_{10}E_{7}^{2}
38 23  a_{9}e_{7}e_{6}a_{1}^{2}  2  aecd D_{10}E_{7}^{2}
38 23  a_{14}e_{6}a_{2}a_{1}^{2}  2  bfaa  A_{17}E_{7}
38 23  a_{11}e_{7}a_{5}a_{1}^{2}  2  cbba  A_{17}E_{7}
38 24  a_{11}d_{9}a_{3}  2  eeb  A_{15}D_{9}  D_{12}^{2}
38 24  a_{12}a_{11}  2  af  A_{12}^{2}  A_{24}
38 25  d_{9}d_{7}a_{7}  1  cdb
38 25  a_{11}d_{8}d_{5}  1  dbc
40 20  a_{15}d_{5}d_{5}  2  bba  A_{15}D_{9}
40 22  d_{8}e_{7}d_{6}a_{3}a_{1}  1  bbada D_{10}E_{7}^{2}
40 22  d_{10}d_{6}^{2}a_{3}  2  bbd D_{10}E_{7}^{2}
40 22  a_{15}d_{6}a_{3}  2  bca  A_{17}E_{7}
40 24  d_{10}d_{6}^{2}a_{1}^{2}  2  bba D_{10}E_{7}^{2}  D_{12}^{2}
40 24E  a_{12}^{2}  4  a  2A_{12}^{2}
40 25  d_{10}a_{9}a_{5}  1  dbb
40 25  a_{15}d_{5}a_{4}  1  bca
40 25  e_{7}e_{6}^{2}a_{5}  2  aaa
40 25  a_{9}e_{7}d_{7}  1  aca
40 25  a_{11}e_{7}d_{5}a_{1}  1  aeba
40 25  a_{14}a_{9}  1  ac
41 24  a_{15}a_{8}  2  ac  A_{15}D_{9}  A_{24}
42 21  a_{13}e_{7}d_{4}  2  aba  A_{17}E_{7}
43 24  a_{16}a_{7}  2  fa  A_{17}E_{7}  A_{24}
43 24E  d_{8}^{3}  6  d  2D_{8}^{3}
44 16  d_{9}d_{8}^{2}  2  db  D_{8}^{3}
44 20  e_{7}^{2}d_{6}d_{5}  2  bad D_{10}E_{7}^{2}
44 24  d_{8}^{2}d_{8}  2  ba  D_{8}^{3} D_{16}E_{8}
44 24  d_{8}^{3}  6  a  D_{8}^{3}  E_{8}^{3}
46 23  d_{11}a_{11}a_{1}^{2}  2  ebd  D_{12}^{2}
46 24  a_{15}d_{8}  2  cb  A_{15}D_{9} D_{16}E_{8}
46 25  d_{11}a_{7}e_{6}  1  dab

47 25  a_{16}d_{7}  1  ca
48 18  d_{10}e_{7}d_{7}a_{1}  1  abda D_{10}E_{7}^{2}
48 18  a_{17}d_{7}a_{1}  2  cac  A_{17}E_{7}
48 22  d_{10}^{2}a_{3}a_{1}^{2}  2  bdb  D_{12}^{2}
48 24  a_{15}e_{7}a_{1}  2  bcc  A_{17}E_{7} D_{16}E_{8}
48 24  d_{10}e_{7}d_{6}a_{1}  1  abaa D_{10}E_{7}^{2} D_{16}E_{8}
48 24  d_{8}e_{7}^{2}a_{1}^{2}  2  aaa D_{10}E_{7}^{2} D_{16}E_{8}
48 25  a_{13}d_{10}a_{1}  1  bcb
49 24E  a_{15}d_{9}  2  aa 2A_{15}D_{9}
50 15  a_{15}d_{10}  2  ba  A_{15}D_{9}
52 20  d_{12}d_{8}d_{5}  1  bad  D_{12}^{2}
52 23  a_{19}a_{4}a_{1}^{2}  2  caa  A_{24}
52 24  d_{12}d_{8}d_{4}  1  bab  D_{12}^{2} D_{16}E_{8}
55 24E  d_{10}e_{7}^{2}  2  dd 2D_{10}E_{7}^{2}
55 24E  a_{17}e_{7}  2  aa 2A_{17}E_{7}
55 25  a_{19}d_{5}  1  aa
56 14  d_{11}e_{7}^{2}  2  da D_{10}E_{7}^{2}
56 21  a_{20}d_{4}  2  aa  A_{24}
58 25  e_{8}a_{11}e_{6}  1  aac
58 25  a_{11}d_{13}  1  bb
60 24  e_{8}d_{8}^{2}  2  aa  E_{8}^{3} D_{16}E_{8}
62 23  a_{15}e_{8}a_{1}^{2}  2  cbd  D_{16}E_{8}
64 22  e_{8}e_{7}^{2}a_{3}  2  aae  E_{8}^{3}
64 22  d_{14}e_{7}a_{3}a_{1}  1  abda  D_{16}E_{8}
67 24E  d_{12}^{2}  2  d  2D_{12}^{2}
68 12  d_{13}d_{12}  1  db  D_{12}^{2}
68 20  d_{12}e_{8}d_{5}  1  aad  D_{16}E_{8}
68 24  d_{12}^{2}  2  b  D_{12}^{2}  D_{24}
71 24  a_{23}  2  a  A_{24}  D_{24}

76 16E  d_{16}d_{9}  1  bd  D_{16}E_{8}
76 16E  d_{9}e_{8}^{2}  2  ea  E_{8}^{3}
76 24  d_{16}d_{8}  1  ba  D_{16}E_{8}  D_{24}
76 24E  a_{24}  2  a  2A_{24}
91 24E  e_{8}^{3}  6  e  2E_{8}^{3}
91 24E  d_{16}e_{8}  1  dd 2D_{16}E_{8}
92 8E  d_{17}e_{8}  1  da  D_{16}E_{8}
100 20  d_{20}d_{5}  1  ad  D_{24}
139 24E  d_{24}  1  d  2D_{24}
140 0E  d_{25}  1  d  D_{24}

References.

[A] Allcock, Daniel. Recognizing equivalence of vectors in the Leech lattice. Preprint

1996, available from http://www.math.utah.edu/~allcock

[B-V] Bacher, R.; Venkov, B. B. Reseaux entiers unimodulaires sans racines en dimension 27 et 28, Preprint 322, Inst. Fourier, Grenoble 1996. Available from

http://www-fourier.ujf-grenoble.fr/PREP/html/a332/a332.html

[B] Borcherds, Richard E. The Leech lattice and other lattices, 1985 PhD thesis, preprint

math.NT/9911195

[B85] Borcherds, R. E. The Leech lattice. Proc. Roy. Soc. London Ser. A 398 (1985), no. 1815, 365–376.

[B90] Borcherds, Richard E. Lattices like the Leech lattice. J. Algebra 130 (1990), no. 1, 219–234.

[C85] Conway, J. H. The automorphism group of the 26-dimensional even unimodular

Lorentzian lattice. J. Algebra 80 (1983), no. 1, 159–163 [C-P-S] Conway, J. H.; Parker, R. A.; Sloane, N. J. A. The covering radius of the Leech lattice.

Proc. Roy. Soc. London Ser. A 380 (1982), no. 1779, 261–290 [C-S] Conway, J. H.; Sloane, N. J. A. Sphere packings, lattices and groups. Third edition.

Grundlehren der Mathematischen Wissenschaften 290. Springer-Verlag, New York, 1999. ISBN: 0-387-98585-9

[C-S98] Conway, J. H.; Sloane, N. J. A. A Note on Optimal Unimodular Lattices J. Number

Theory, 72 (1998), pp. 357–362.

[E] Eichler, M. “Quadratische Formen und Orthogonal gruppen”, Springer–Verlag, 1952. [E-G] Elkies, Noam D.; Gross, Benedict H. The exceptional cone and the Leech lattice.

Internat. Math. Res. Notices 1996, no. 14, 665–698.

[K] Kneser, Martin. Klassenzahlen definiter quadratischer Formen. Arch. Math. 8 (1957), 241–250.

[N] Niemeier, H.-V. Definite quadratische Formen der Dimension 24 und Diskriminante 1, J. Number theory 5 (1973), 142–178.

[S-V] Scharlau, Rudolf; Venkov, Boris B. The genus of the Barnes-Wall lattice. Comment.

Math. Helv. 69 (1994), no. 2, 322–333.

[V75] Vinberg, È. B. Some arithmetical discrete groups in Lobačevskiĭ spaces. Discrete subgroups of Lie groups and applications to moduli (Internat. Colloq., Bombay, 1973), pp. 323–348. Oxford Univ. Press, Bombay, 1975.

[W] Watson, G. N. The problem of the square pyramid. Messenger Math. 48 (1919), 1-22.