5.1 Review of the visible root system of a neighbor

For $x \in {\mathbb {R}}^n$ and $j \in {\mathbb {R}}$ , we denote by $\mathrm {m}_j(x) \in \{0,1,\dots ,n\}$ the number of coordinates of x which are equal to j. A pair $(d,x)$ , with $d\geq 1$ an integer and $x \in {\mathbb {Z}}^n$ , will be said to be normalized if we have $1=x_1 \leq x_2 \leq \cdots \leq x_n \leq d/2$ , as well as $\mathrm {m}_1(x) \geq \mathrm {m}_j(x)$ for all $1 \leq j <d/2$ . We call $\mathrm {m}_1(x)$ the index of $(d,x)$ , and $\mathrm {m}_{d/2}(x)$ the end Footnote ¹³ of $(d,x)$ . Finally, the type of x, or of $(d,x)$ , is the partition of the integer n defined by the nonzero integers $\mathrm {m}_j(x)$ , $j \in {\mathbb {Z}}$ .

As the isometry group of the lattice $\mathrm {I}_n={\mathbb {Z}}^n$ consists of the $2^n n!$ permutations and sign changes of coordinates, we deduce the following:

Assume now $x \in {\mathbb {Z}}^n$ is d-isotropic and $(d,x)$ is normalized. Recall that the visible root system of the d-neighbor $N:=\mathrm {N}_d(x;\epsilon )$ of $\mathrm { I}_n$ is defined in [Reference Chenevier6] §5 as

$$ \begin{align*}R^{\mathrm{v}}:=\mathrm{R}_2(N) \cap \mathrm{I}_n=\mathrm{R}_2(\mathrm{ M}_d(x)).\end{align*} $$

As explained in §5.1 loc. cit., it is straightforward to determine $R^{\mathrm {v}}$ from an inspection of the equal coordinates of $x \bmod d{\mathbb {Z}}^n$ : if $(d,x)$ has end e and type $n_1+\dots +n_s+e$ , then we have

(5.1) $$ \begin{align} R^{\mathrm{v}} \simeq \mathbf{A}_{n_1-1} \mathbf{A}_{n_2-1} \cdots \mathbf{A}_{n_s-1} \mathbf{D}_e. \end{align} $$

The actual root system R of N is the union of $R^{\mathrm {v}}$ and of the (typically more mysterious) roots belonging to the other $\mathrm {M}_d(x)$ -cosets of N generated by the multiples of the ‘glue vector’ $\frac {x'}{d}$ of Formula 1.1. Several constraints force $R^{\mathrm {v}}$ to be different from R in general. For instance, as Formula (5.1) shows, $R^{\mathrm {v}}$ cannot contain any irreducible component of type $\mathbf {E}$ , has at most one component of type $\mathbf {D}$ , and has a limitation for its type $\mathbf {A}$ components as well (for instance, it has at most $n/2$ components of type $\mathbf {A}_1$ ). Another, more subtle, constraint is that $R^{\mathrm {v}}$ tends to generate a saturated sublattice in N (this holds at least when d is a large prime), which is not the case for R in general. We refer to loc. cit. Section 5 for a detailed study all of those constraints on $R^{\mathrm {v}}$ .

5.2 The BNE algorithm

We now describe in detail the Biased Neighbor Enumeration algorithm, later refered to as BNE, informally discussed in [Reference Chenevier6, §1.10]. It takes the following as inputs: an integer $n \geq 1$ , an isomorphism class R of root systems, the reduced mass $\rho $ of $\mathrm {X}_n^R$ , an integer $0\leq e<n$ and an integer partition $n-e=\sum _{i=1}^s n_i$ , with $n_1 \geq n_2 \geq \cdots \geq n_s\geq 1$ . If BNE terminates, it returns a list of $(d,x,\epsilon ,\mu ,\beta )$ where the lattices $\mathrm {N}_d(x; \epsilon )$ are representatives of $\mathrm {X}_n^R$ , and where $\mathrm { N}_d(x;\epsilon )$ has reduced mass $\mu $ and BV invariant $\beta $ ; it also proves that all the elements in $\mathrm {X}_n^R$ have distinct $\mathrm {BV}$ invariants.

1. 1. Set $d=2$ and define empty lists $inv$ and $lat$ .

2. 2. Make the list L of all $x \in {\mathbb {Z}}^n$ with $(d,x)$ normalized of type $n_1+\dots +n_s+e$ , index $n_1$ and end e.

3. 3. Only keep in L those x such that x is d-isotropic.

4. 4. For each $x \in L$ andFootnote ¹⁵ $\epsilon \in \{0,1\}$ , compute the cyclic d-neighbor $\mathrm {N}_d(x; \epsilon )$ of $\mathrm {I}_{n}$ .

5. 5. Replace L with the list of $(x,\epsilon ) \in L \times \{0,1\}$ , satisfying both $\mathrm {r}_1(\mathrm {N}_d(x;\epsilon ))=0$ and $\mathrm {R}_2(\mathrm {N}_d(x;\epsilon )) \simeq R$ .

6. 6. Compute the (hashed) $\mathrm {BV}$ invariants $\beta $ of all $\mathrm {N}_d(x;\epsilon )$ with $(x,\epsilon ) \in L$ .

7. 7. For each $(x,\epsilon ) \in L$ , if the BV invariant $\beta $ of $\mathrm {N}_d(x;\epsilon )$ is not in $inv$ , do the following:

   1. 7a. Add $\beta $ to the list $inv$ ,

   2. 7b. Compute the reduced mass $\mu $ of $\mathrm {N}_d(x;\epsilon )$ ,

   3. 7c. Add $(d,x,\epsilon ,\beta ,\mu )$ to the list $lat$ ,

   4. 7d. Set $\rho \leftarrow \rho -\mu $ . If $\rho =0$ , return $lat$ .

8. 8. $d \leftarrow d+1$ and go back to Step 2.

Let us first state an important criterion for BNE to terminate. We attach to e and the integer partition $n=n_1+\cdots +n_s+e$ the isomorphism class of root system

$$ \begin{align*}V:= \mathbf{A}_{n_1-1}\mathbf{A}_{n_2-1} \cdots \mathbf{ A}_{n_s-1}\mathbf{D}_e.\end{align*} $$

It follows from Formula (5.1) that each $\mathrm {N}_d(x;\epsilon )$ at Step 4 of BNE has a visible root system isometric to V. For an ADE root system S, and an integral lattice L, we denote by $\mathrm {emb}(S,L)$ the number of isometric embeddingsFootnote ¹⁶ $\mathrm {Q}(S) \rightarrow L$ with saturated image. Following §5.9 loc. cit., we say that a pair $(R,S)$ of ADE root systems is safe if for any integral lattices L with $\mathrm {r}_1(L)=0$ and $\mathrm {R}_2(L) \simeq R$ , we have $\mathrm { emb}(S,L) \neq 0$ .

Let us explain how this theorem follows from Theorem E of [Reference Chenevier6] (a special case of the results of [Reference Chenevier5]). By Fact 5.2 and Formula (5.1), Steps 2 and 3 enumerate representatives $(d,x,\epsilon )$ for almost all $\mathrm {O}(\mathrm {I}_n)$ -orbits of the cyclic d-neighbors $\mathrm {N}_d(x;\epsilon )$ of $\mathrm {I}_n$ whose visible root system is isomorphic to V. In the case d is prime, all orbits are considered in these two steps. By the aforementioned Theorem E loc. cit., for any given $L \in \mathrm {X}_n$ , the proportion of d-neighbors of $\mathrm {I}_n$ having a given visible root system V, and which are isomorphic to L, tends to (an absolute constant times) $\mathrm {emb}(V,L)/|\mathrm {O}(L)|$ when the prime d goes to $\infty $ . This number is nonzero when L is in $\mathrm {X}^R_n$ and $(R,V)$ is safe, which shows that BNE terminates assuming $\mathrm {BV}$ is sharp – hence Theorem 5.3. We stress, however, that there is no known upper bound on the integer d such that each $L \in \mathcal {L}_n$ is isometric to a d-neighbor of $\mathrm {I}_n$ . This method for searching for unimodular lattices is probabilistic and a form of (non-uniform) coupon collector problem: see [Reference Chenevier6, §1.12] for a discussion along these lines.

Of course, the $\texttt {BNE}$ algorithm is interesting even if it does not terminate, since it usually finds many lattices if not all. We finally give the non-biased version of BNE.

5.3 Further remarks and details on the various steps of BNE

Remark 5.6 (Step $2'$ : lines versus vectors, and redundancies)

Let $x,x' \in {\mathbb {Z}}^n$ be two d-isotropic vectors, say with $(d,x)$ and $(d,x')$ normalized of same type $n_1+\dots +n_s+e$ . Denote by $\ell $ and $\ell ' \subset \mathrm {I}_n \otimes {\mathbb {Z}}/d$ the ${\mathbb {Z}}/d$ -line they generate, and say that $(d,x)$ and $(d,x')$ are equivalent if $\ell $ and $\ell '$ are in the same $\mathrm {O}(\mathrm {I}_n)$ -orbit. The d-neighbors defined by equivalent $(d,x)$ and $(d,x')$ are naturally isomorphic, creating unwanted redundancy in our algorithm. For a given $(d,x)$ , then there are exactly f equivalent pairs $(d,x')$ , with

$$ \begin{align*} f:=|\,\{\,\,1\leq i \leq d/2\,\,\,|\,\,\,\mathrm{m}_i(x)=\mathrm{m}_1(x)\,\,\& \,\,\mathrm{gcd}(i,d)=1\,\}\,|. \end{align*} $$

This is especially large in the case $n_i=1$ for all i, and d is prime, for which we have $f=n$ . We did not find any clever way to directly select a single element among those f ones, but did instead the following right after Step 2 in the algorithm: fix some total ordering $\prec $ on ${\mathbb {Z}}^n$ , and for each $(d,x) \in L$ , compute all equivalent $(d,x') \in L$ , and only keep $(d,x)$ in L if it is the biggest of them for $\prec $ .

6.1 The root system $7\mathbf {A}_1\,3\mathbf {A}_2\,\mathbf {A}_7$ in dimension $28$

For this root system R, we know from King that the reduced mass of $\mathrm {X}_{28}^R$ is $5/24$ (whereas its mass is $|\mathrm {W}(R)|=1\,114\,767\,360$ times smaller), so we expect $\mathrm {X}_{28}^R$ to contain only very few lattices. We choose the visible root system $V:=6\mathbf {A}_1\,2\mathbf {A}_2\,\mathbf {A}_7$ , that is $e=0$ and the integer partition $8\,3^2\,2^6\,1^2$ of $28$ . We have $s=11$ , so we may start at $d=23$ . The $\texttt {BNE}$ algorithm terminates at $d=27$ and returns the $3$ lattices in $\mathrm {X}_{28}^R$ , with reduced masses $1/12$ , $1/12$ and $1/24$ , after about $7$ minutes of CPU time.

We will give some details describing intermediate steps of such calculations using tables such as Table 8. The first column contains d. The second column gives the number $\# \texttt {iso}$ of d-isotropic lines found after Steps $2$ and $3$ , incorporating Step $2'$ explained in Remark 5.6. The third column gives the size $\#\texttt {found}$ of the list L after Step $5$ . The better we have chosen the visible root system V, the higher the ratio $\# \texttt {iso}/\#\texttt {found}$ is. The fourth column gives the number of new lattices found in L, and the last column, the remaining reduced mass $\rho $ of $\mathrm {X}_{n}^R$ after Step $7$ : when $0$ , the algorithm terminates.

Table 8

Hunting $\mathrm {X}_{28}^R$ with $R=7\mathbf {A}_1\,3\mathbf {A}_2\,\mathbf {A}_7$ and $V=6\mathbf {A}_1\,2\mathbf {A}_2\,\mathbf {A}_7$

6.2 The root system $4\mathbf {A}_1\,2\mathbf {A}_2 \,2\mathbf {A}_3\,\mathbf {D}_4$ in dimension $28$

As another example with a little more lattices, as well as a component of type $\mathbf {D}$ , consider the case of the root system R of the title. The reduced mass of $\mathrm {X}_{28}^R$ is $1033/16$ . We use the visible root system $V=3\mathbf {A}_1\,2\mathbf {A}_2 \,2\mathbf {A}_3\,\mathbf {D}_4$ , that is $e=4$ and type $4^3\,3^2\,2^3\,1^4$ (hence, d is even $\geq 24$ ). The BNE algorithm terminates at $d=36$ . It returns the $156$ lattices of $\mathrm {X}_{28}^R$ , with reduced mass $1/2$ ( $112$ times), $1/4$ ( $26$ times), $1/8$ ( $15$ times) and $1/16$ ( $3$ times), after about $24$ h of CPU time: see Table 9.

Table 9

Hunting $\mathrm {X}_{28}^R$ with $R=4\mathbf {A}_1\,2\mathbf {A}_2 \,2\mathbf {A}_3\,\mathbf { D}_4$ and $V=3\mathbf {A}_1\,2\mathbf {A}_2 \,2\mathbf {A}_3\,\mathbf { D}_4$

6.3 The root system $16\mathbf {A}_1\, \mathbf {E}_6$ in dimension $28$

For this root system R, the reduced mass of $\mathrm {X}_{28}^R$ is $1/23040$ ; hence, there may well be a unique class in $\mathrm {X}_{28}^R$ . The difficulty here is that no possible visible root system is ‘very close’ to R, because of the presence of a component of type $\mathbf {E}$ and too many $\mathbf {A}_1$ . The best idea here is to use the fact that $\mathbf {A}_5 \mathbf {A}_1$ is a $2$ -kernel of $\mathbf {E}_6$ (see Example 5.22 in [Reference Chenevier6]) and to run BNE for the visible root system $V \simeq 10\mathbf { A}_1\,\mathbf {A}_5\,\mathbf {D}_2 \simeq 12\mathbf {A}_1\,\mathbf {A}_5$ ; hence, $e=2$ and type $6\,2^{11}$ (this forces d even and $\geq 24$ ). For each of the even integers $24 \leq d \leq 46$ , it turns out that the values of $\#\texttt {iso}$ given by BNE are

$$ \begin{align*}0, 0, 4, 5, 42, 93, 344, 516, 1440, 2064, 5792, 7673,\end{align*} $$

but that in all cases, we have $\#\texttt {found}=0$ . Nevertheless, for $d=48$ , we find $\#\texttt {iso}=17098$ and finally $\#\texttt {found}=1$ ! The found (and unique) lattice in $\mathrm {X}_{28}^R$ has reduced mass $1/23040$ and is $\mathrm {N}_{48}(x; 0)$ with

The whole computation only takes about $13$ minutes of CPU time (essentially because we computed a single $\mathrm {BV}$ invariant). The explanation for all those zeros values of $\#\texttt {found}$ above is that this choice of visible root system, although essentially the best we can take, will most likely find lattices with root system of the form $n\mathbf { A}_1 \mathbf {A}_5$ with $n\geq 13$ . Indeed, there are many lattices in $\mathrm {X}_{28}$ with root systems $13\mathbf {A}_1 \mathbf {A}_5$ (including some with the large reduced mass $1/4$ ) and $17\mathbf {A}_1\mathbf {A}_5$ .

Remark 6.1. The root system $\mathbf {D}_5$ is also a $2$ -kernel of $\mathbf {E}_6$ , and the lattice L above can also be found using the visible root system $11 \mathbf {A}_1 \mathbf {D}_5$ for $d=50$ (within about 55 minutes). In the latter case, we have $\#\texttt {iso}=31790$ , $\#\texttt {found}=7$ , and obtained $L \simeq \mathrm {N}_{50}(y)$ with $y \in {\mathbb {Z}}^{28}$ defined as

All other choices of visible root systems seem to require more computations.

6.4 The root system $8\mathbf {A}_1\,2\mathbf {A}_2$ in dimension $28$

For this root system R, the reduced mass of $\mathrm {X}_{28}^R$ is $9694663/2880$ $\simeq 3366.2$ , which already implies $|\mathrm {X}_{28}^R|\geq 6733$ . Actually, as we shall see, we have $|\mathrm { X}_{28}^R|=7603$ , and this is actually the maximum of the $|\mathrm {X}_{28}^S|$ for all root systems S. This is one reason why we chosed this example, which is much harder than the previous ones.

Table 10

Hunting $\mathrm {X}_{28}^R$ with $R=8\mathbf {A}_1\,2\mathbf {A}_2$ and $V=7\mathbf { A}_1\,2\mathbf {A}_2$

We know from Section 5.9 [Reference Chenevier6] (and especially Example 5.14) that the pair $(R,V)$ is safe for $V=7\mathbf {A}_1\,2\mathbf {A}_2$ , so we run $\texttt {BNE}$ for this visible root system, that is $e\leq 1$ and type $3^2\,2^7\,1^8$ ( $s=17$ ). For the subsequent purpose of computing $2$ -neighbors, it is convenient to restrict our search to odd d. Table 10 gives a summary of what $\texttt {BNE}$ finds for all odd d from $2s+1=35$ to $43$ (we add in the table a lower bound of the number $\# \texttt {rem_lat}$ of remaining lattices in $\mathrm {X}_{28}^R$ after Step 7 in the last column). This computation is already quite lengthy: it took about $1400$ h of CPU time. It allowed to find $7506$ different classes in $\mathrm {X}_{28}^R$ so far, with remaining reduced mass $899/36$ . It would be natural to try to go on in order to find the remaining lattices – namely, by exploring $d=45$ , $47$ and so on…This would unfortunately require much more CPU time. Instead, we may proceed as follows.

(a) Exceptional lattices in $\mathrm {X}_{28}^R$ . An inspection of the $7506$ found lattices is that none of them is exceptional. We refer to Lemma 6.3 for an explanation of an analogous phenomenon for $\mathrm { X}_{29}^\emptyset $ . Our aim now is to determine the (classes) of exceptional unimodular lattices in $\mathrm {X}_{28}^R$ .

As we are in dimension $28 \equiv 4 \bmod 8$ , exceptional lattices L in $\mathcal {L}_{28}$ with $\mathrm { r}_1(L)=0$ may be described using their singular companion $L':=\mathrm {sing}(L)$ , by Proposition 2.1. Write $L' \simeq \mathrm {I}_r \perp U$ with $r\geq 1$ and $U \in \mathcal {L}_{28-r}$ satisfying $\mathrm { r}_1(U)=0$ . In particular, $\mathbf {D}_r \simeq \mathrm {R}_2(\mathrm {I}_r)$ is an irreducible component of $\mathrm { R}_2(L')=\mathrm {R}_2(L)$ (recall $\mathbf {D}_1=\emptyset $ and $\mathbf {D}_2\,=\,2\mathbf {A}_1$ ). Going back to the root system $R \simeq 8\mathbf {A}_1\,2\mathbf {A}_2$ , the only possibilities are thus either $r=1$ and $\mathrm {R}_2(U) \simeq R$ , or $r=2$ and $\mathrm {R}_2(U) \simeq 6\mathbf {A}_1\,2\mathbf {A}_2$ . But an inspection of the classification in [Reference Chenevier6] of $\mathrm {X}_{26}$ and $\mathrm {X}_{27}$ shows that there are exactly two possibilities for U for $r=2$ , and $89$ for $r=1$ (all being nonexceptional). Proposition 2.1 thus shows the following:

Better, in the analysis above, if we have $U \,\simeq \,\mathrm {N}_d(x)$ with d odd and $x \in {\mathbb {Z}}^{28-r}$ , then we know from Lemma 11.1 and Proposition 11.6 in [Reference Chenevier6] that the corresponding L satisfies $L \simeq \mathrm {N}_{2d}(y;0)$ where $y \in {\mathbb {Z}}^{28}$ has odd coordinates and satisfies $y_i \equiv x_i \bmod d$ for $i < 28-r$ , and $y_i =d$ for $i\geq 28-r$ . Neighbor forms for the $2+89$ different U are easily found with d odd using $\texttt {BNE}$ in dimensions $26$ and $27$ . For instance, the $2$ classes U in $\mathrm { X}_{26}$ with root system $6\mathbf {A}_1\, 2\mathbf {A}_2$ are those of the $\mathrm {N}_{35}(x)$ with $x \in {\mathbb {Z}}^{26}$ given by

and their respective reduced mass is $1/288$ and $1/24$ . In the end, we do obtain neighbor forms for all $91$ exceptional lattices of Lemma 6.2.

(b) Strict $2$ neighbors of rare lattices. After incorporating the $91$ exceptional lattices above in $\mathrm {X}_{28}^R$ , the remaining reduced mass is $13/96$ . The remaining lattices presumably all have a small reduced mass. In order to find them, we apply some ideas from the theory of visible isometries explained in [Reference Chenevier6, §7] (see especially §7.9):

(b1) The first idea is that if some $L \in \mathcal {L}_n$ has a large isometry group, we expect that many $2$ -isotropic lines in $L/2L$ will be stable by a nontrivial isometry of L, hence producing a $2$ -neighbor having that isometry, and biasing the search.

(b2) A second idea is that in order to avoid computing all $2$ -neighbors of L (or $\mathrm {O}(L)$ -orbits of them), it seems more promising to focus on those having the same visible root system as L viewed as neighbors of $\mathrm {I}_n$ ; we call them the strict $2$ -neighbors of L. Concretely, if we have $L=\mathrm {N}_d(x)$ with d odd, $x \in {\mathbb {Z}}^n$ and $(d,x)$ normalized, then the strict $2$ -neighbors of L are the $\mathrm {N}_{2d}(y;\epsilon )$ with $y \in {\mathbb {Z}}^{n}$ satisfying $y \equiv x \bmod d$ , as well as $y_1=1$ and, for all $1\leq i,j \leq n$ with $x_i = x_j$ , the congruence $y_i \equiv y_j \bmod 2$ .

Among the $494$ lattices found above for $d=41$ and $d=43$ , there are $12$ lattices with reduced mass $\leq 1/32$ . For each of these d-neighbors L, given as $L=\mathrm {N}_d(x)$ , we compute all the strict $2$ -neighbors $L'$ of L, with $L'=\mathrm {N}_{2d}(y,\epsilon )$ as in (b2) above. As we are in dimension $n=28$ and L has a visible root system $7\mathbf {A}_1\,2\mathbf {A}_2$ , there are thus only $2^{28-2-7-4-1}=65\,536$ choices for y, hence presumably $32\,768$ isotropic ones, a quite manageable quantity. The hope is to find the remaining lattices among those $2\times 12\times 32\,768=786\,432$ different $2$ -neighbors. It works! Indeed, in about $20$ h of CPU time, we did find this way the $6$ remaining elements of $\mathrm {X}_{28}^R$ . They have a reduced mass $1/96$ , $1/64$ twice and $1/32$ three times, and are given by the values of $(d,y,\epsilon )$ in Table 11.

Table 11

The last $\mathrm {N}_{2d}(y;\epsilon )$ found in $\mathrm {X}_{28}^R$ for $R=8\mathbf { A}_1\,2\mathbf {A}_2$

Table 12

Hunting $\mathrm {X}_{29}^\emptyset $ with $V=\emptyset $ and odd $59 \leq d\leq 83$

6.5 The empty root system in dimension $29$

We finally discuss the determination of $\mathrm {X}_{29}^\emptyset $ . For such lattices, the mass and the reduced mass coincide so we usually omit the term ‘reduced’. We know that the mass of $\mathrm {X}_{29}^\emptyset $ is $49612728929/11136000$ $\simeq 4455.2$ , so we have at least $8911$ isometry classes. The only possibility here is to take $V=\emptyset $ , and we choose $0 \leq e \leq 1$ and all $n_i$ equal to $1$ . See Table 12 for a summary of what $\texttt {BNE}$ finds for all odd d from $2s+1=59$ to $83$ , after about $850$ h of CPU time. At this step, we have found $9964$ elements in $\mathrm {X}_{29}^\emptyset $ . For instance, the first of theses lattices, found for $d=59$ , is the lattice $\mathrm {N}_{59}(1,2,3,\dots ,29)$ which incidentally belongs to the family studied in [Reference Chenevier6, §8]. However, an inspection of our list shows that we only found a single exceptional lattice so far, for $d=83$ . This can be partially explained by the following lemma.

Note that for $n=29$ and $\xi .\xi =5$ , this forces $p\geq 83$ , in accordance with what we found. Our first aim now is to seek for exceptional lattices in $\mathrm {X}_{29}^\emptyset $ . We cannot argue as in § 6.4 (a) since $29 \not \equiv 4 \bmod 8$ . Instead, we use the variant of BNE discussed in [Reference Chenevier6, §9.13–9.16]. The basic idea is to look for d-neighbors $N:=\mathrm {N}_d(x;\epsilon )$ of $\mathrm {I}_{29}$ with empty root systems and such that the norm $5$ vector $(0,\dots ,0,1,1,1,1,1) \in \mathrm {I}_{29}$ is a (visible!) characteristic vector of N. As explained loc. cit., the trick is just to modify Step 2 of BNE and enumerate only d-isotropic $x \in {\mathbb {Z}}^{29}$ , with d even, all of whose coordinates are odd, except the last $5$ ones which are even and with sum $\equiv 0 \bmod d$ . This forces $d\geq 94$ , and Table 13 shows what we obtain after only about $60$ h of CPU time. The last lattice, with mass $1/24000$ , came very late and is much harder to find than the others: this is $\mathrm {N}_{114}(x;0)$ with

Up to this point, we have found $10068$ classes in $\mathrm {X}_{29}^\emptyset $ , with remaining mass $1481/5760$ . There are several methods that we can use to find the last ones. First, we run BNE for the empty visible root system, $0\leq e \leq 1$ , and even $58 \leq d \leq 82$ , which we have not done yet, and only finds $15$ more lattices in about $1050$ more hours, as shown by Table 14.

Table 13

Hunting exceptional lattices in $\mathrm {X}_{29}^\emptyset $

Table 14

Hunting $\mathrm {X}_{29}^\emptyset $ with $V=\emptyset $ and even $58 \leq d\leq 82$

At this point, we have found $10\,083$ lattices, and the remaining mass is only $301/15360$ . It would be presumably quite lengthy to seek for the remaining lattices by simply pursuing BNE. A more direct way to find them is to use the theory of visible isometries as in §6.4 (b), and study $2$ -neighbors of the found lattices $L=\mathrm {N}_d(x)$ with large isometry groups and d odd. Note that as the root system is empty here, each $2$ neighbor of such an L is strict, so we have no meaningful way to reduce the search as we did in §6.4. In such a situation with $\mathrm {O}(L)$ big, we should also gain much in principle in computing first the $\mathrm {O}(L)$ -orbits of $2$ -isotropic lines in $L/2L$ , since there is a huge number of such lines in dimension $29$ . In practice, this is not really necessary, and we prefer to only compute the neighbors associated to a large number of $2$ -isotropic lines of a given L, say here $2\,000\,000$ , and then try the next L if it fails.

This works pretty well! Indeed, the lattice $L=\mathrm {N}_d(x)$ in $\mathrm {X}_{29}^\emptyset $ , with d odd and largest isometry group, appears for $d=83$ and satisfies $|\mathrm {O}(L)|=1536$ . Using this L as explained above, we do find $7$ new lattices, with remaining mass $167/25920$ . The next $3$ lattices with largest isometry groups did not provide new lattices, but the $2$ after – namely, a $75$ -neighbor with mass $1/160$ , and a $81$ -neighbor with mass $1/128$ – do give rise (each) to a new lattice, with respective masses $1/160$ and $1/5184$ , and this concludes the proof! Those $7+1+1=9$ lattices $\mathrm {N}_{2d}(y;\epsilon )$ are given in the following table. The full computation here took about $480$ h of CPU time.

Table 15

The last nine lattices $\mathrm {N}_{2d}(y;\epsilon )$ found in $\mathrm { X}_{29}^\emptyset $

The method above is essentially the one we used when we first computed $\mathrm {X}_{29}^\emptyset $ . Meanwhile, the second author found a more direct way to find d-neighbors of $\mathrm {I}_n$ with prescribed (and ‘visible’) isometries. This method is described in §7.5 and §7.7 of [Reference Chenevier6]. The basic idea is to fix $\sigma \in \mathrm {O}(\mathrm {I}_n)$ and to study to d-neighbors N of $\mathrm { I}_n$ with $\sigma \in \mathrm {O}(N)$ and having a given visible root system V, which translates into some conditions on the isotropic lines that we enumerate. This is especially suited to empty (or small) V. An example of application of these ideas to the determination of $\mathrm {X}_{28}^\emptyset $ is detailed in §7.6 loc. cit. The situation here is a bit similar, and goes as follows.

We go back right before the $2$ -neighbor argument above. At this step, the remaining mass is $301/15360$ . We have $15360 \,=\, 2^{10}\,3\,5$ , so we know that we still need to find lattices in $\mathrm { X}_{29}^\emptyset $ having isometries of prime order $q=2, 3$ and $5$ . The characteristic polynomial of such an isometry is $\phi _q^k \,\phi _1^l$ , with $\phi _m$ the m-th cyclotomic polynomial, and $k(q-1)+l=29$ . For fixed q and k, we choose an auxiliary odd prime $p \equiv 1 \bmod q$ and consider associated $d:=pd'$ isotropic lines described in loc. cit. §7.5 for all odd integers $d'=1,\,3,\,\dots $ and so on. These lines have the properties that the associated neighbors N have an empty visible root system and a (visible) isometry with characteristic polynomial $\phi _q^k \,\phi _1^l$ . Better, they are tailored such that the following product of k disjoint q-cycles in $\mathrm {S}_{29} \subset \mathrm {O}(\mathrm { I}_{29})$ ,

$$ \begin{align*}\sigma_{q,k}\,\,=\,\,(1\,2\,\dots \,\,q)\,\,(q+1\,\,\,q+2\,\,\,\dots\,\,\,2q)\,\, \cdots \,\,(\,(k-1)q+1\,\,\,(k-1)q+2\,\,\,\dots \,\,\,kq),\end{align*} $$

lies in $\mathrm {O}(N)$ . This requires $qk \leq 29$ . By studying those lines, we do also find the $9$ remaining lattices, under the form given by Table 16 below. This is much faster: it only took less than $3$ h to find the first $8$ lattices, and about $7$ h for the last one.

Table 16

Another form for the last nine lattices $\mathrm {N}_d(x)$ found in $\mathrm { X}_{29}^\emptyset $

For instance, for the lattice $L=\mathrm {N}_{329}(x)$ above with mass $1/18432$ , we have

so $\sigma _{3,9}^{-1}$ acts on $({\mathbb {Z}}/7)x \subset ({\mathbb {Z}}/7)^{29}$ by multiplication by $2$ , and fixes $x \bmod 47$ .