The 167889 positive-definite even lattices of rank 18 and discriminant 163

Lattices computed in 2018; posted January 2026 with some further information

• GRAM.gp.bz2: GP vector G of Gram matrices for the 167889 positive-definite even lattices of rank 18 and discriminant 163 (compressed from 112.5M to 9.27M)
• SIG.gp.bz2: GP vector SIG of the signatures [name, r_MW, T_MW, D_R, c, N(L), N(R⁺(L)), #Aut(L), [Aut(L) : W_L]] for those lattices L. Here
  ○ name is a string such as "A1^2A2A3A4" for A₁² A₂ A₃ A₄ (which is the most common root lattice in this genus);
  ○ r_MW is the Mordell–Weil rank, equal to 18 minus the rank of R(L);
  ○ T_MW is the structure of the Mordell–Weil torsion group, isomorphic to R⁺(L) / R(L), and for each of our lattices equal to one of [] or [2] or [3] or [4]  (it would be [6,2] for a group Z/2Z × Z/6Z);
  ○ D_R is the discriminant of R(L);
  ○ c is the number of simple components of R(L) (for example, 5 for each of the 3650 lattices with R(L) = A₁² A₂ A₃ A₄);
  ○ N(L) is a vector of length 3 containing the counts of vectors of norm 2, 4, 6 in L;
  ○ N(R⁺(L)) is a vector of length 2 containing the counts of vectors of norm 4 and 6 in R⁺(L) (this can distinguish between non-isomorphic lattices R⁺(L) that have the same R(L) and R⁺(L) / R(L)); note that we do not include norm 2 because that count is the same for L and R⁺(L);
  ○ #Aut(L) is the size of the group of automorphisms of L, e.g. 138240 = 2 · 2!² 3! 4! 5! for each of the lattices with R(L) = A₁² A₂ A₃ A₄)); and finally
  ○ the index [#Aut(L) : W_L], which is always even (because R(L) cannot have rank 18 so W_L cannot contain the central involution −1), and is usually 2 (including for all the A₁² A₂ A₃ A₄ lattices) but sometimes equals 4 or 6.
• M.gp: GP vector M of Gram matrices for the 16 positive-definite even lattices of rank 19 and discriminant 4, from M[1] = D₁₉ to M[16] = (A₇²D₅)⁺⁸
  (Exception: for M[1] we use matid(19) = the 19×19 identity matrix, which is a Gram matrix for the discriminant-1 lattice Z¹⁹⊃D₁₉ — see the next item);
• V.gp.bz2: GP vector V = [V[1], V[2], ..., V[16]] whose k-th entry is a list of integer vectors c of length 19 that represent the hyperplane slices of M_k that give the k-th batch of lattices L. The encoding is as follows. Suppose G_k is the Gram matrix of M_k corresonding to the Z-basis e₁, … e₁₉. If v* is a dual vector (of norm 163/4 in our setting) for M_k then the hyperplane orthogonal to v* consists of combinations ∑_i a_i e_i  whose coefficients a_i satisfy ∑_i c_i a_i = 0, where each c_i is the integer ⟨v*, e_i⟩. These c_i are the entries of c. Hence the following GP code gives a Gram matrix for L:

    K = matkerint(Mat(c));
    K~ * M[k] * K  

  If M is the integral lattice of discriminant 1 whose even sublattice is M then we get the same hyperplane slice by replacing M by M and v* by 2v* which is a characteristic vector for M. In most cases we preferred to work with M rather than M, but for k=1 the lattice M associated to M = D₁₉ is simply Z¹⁹, offering a shortcut: c is simply one of the 22 vectors of norm 163 consisting of 19 odd integers in non-decreasing order, and then  K~ * M[k] * K  simplifies to  K~ * K .

  The file GRAM.gp can be recovered from M.gp and V.gp using the following GP code, followed by removing spaces to reduce the file size, inserting line breaks for better legibility, prepending the introductory "allocatemem(2^32)" and "{G = ", and appending "};":

    G = [];
    {
    for(k=1,16,
      G_k = vector(#V[k]);
      for(n=1,#V[k],
        K = matkerint(Mat(V[k][n]));
        G_k[n] = K~ * M[k] * K
      );
      G = concat(G,G_k)
    )
    }
  
    print(G)
      

  (This is GP Version 2.17.1 .  I don’t know whether other versions are guaranteed to have the same matkerint output; if not, the results may be different, but will still be Gram matrices for the same 167889 lattices.)