{
 "description": "The rank-19 even positive-definite root-free lattices W512, W512', W1024 (Theorem 3.5, Eq. (34) and Table 3 of the main text; SM Section S5.1, Eqs. (S5.1)-(S5.3), Table S5) and Lambda_19 (SM Section S1.2 and SM Eq. (S5.4)), with the primitive embeddings W(-1) -> Gamma_{3,19} of SM Eqs. (S5.5)-(S5.7). Conventions: Gram matrices are lists of integer rows; kissing_number counts v and -v; discriminant_form gives the invariants of L^*/L, the values q(x_i) mod 2 of the discriminant quadratic form on generators x_i and the matrix b(x_i,x_j) mod 1. For each gluing, overlattice_gram is the 22x22 Gram matrix G of an even unimodular lattice of signature (3,19); embedding_rows X (19x22) satisfies X G X^T = -gram (the image of W(-1)); complement_basis_rows K (3x22) spans the orthogonal complement, with K G K^T = complement_gram; glue_index = |det [X;K]|, and det(W) det(C) / glue_index^2 = |det G| = 1.",
 "provenance": {
  "W1024": "obtained by Kneser 3-neighbour steps from E8+E8+diag(2,2,c') with c' = 256 (SM Section S5.1)",
  "W512": "obtained by Kneser 3-neighbour steps from E8+E8+diag(2,2,c') with c' = 128 (SM Section S5.1)",
  "W512p": "obtained by Kneser 3-neighbour steps from E8+E8+diag(2,2,c') with c' = 128 (SM Section S5.1)",
  "Lambda19": "orthogonal complement of sqrt(2)D5 in the Leech lattice"
 },
 "lattices": {
  "W512": {
   "rank": 19,
   "gram": [
    [
     4,
     2,
     2,
     1,
     2,
     0,
     2,
     2,
     -2,
     2,
     -2,
     -1,
     -1,
     -2,
     2,
     1,
     2,
     2,
     -1
    ],
    [
     2,
     4,
     2,
     -1,
     2,
     -1,
     2,
     1,
     -2,
     0,
     0,
     -1,
     1,
     0,
     1,
     2,
     1,
     1,
     0
    ],
    [
     2,
     2,
     4,
     1,
     1,
     1,
     2,
     2,
     -2,
     0,
     0,
     -1,
     0,
     -1,
     2,
     0,
     2,
     2,
     -1
    ],
    [
     1,
     -1,
     1,
     4,
     1,
     2,
     1,
     2,
     1,
     2,
     0,
     1,
     -1,
     -2,
     2,
     -1,
     0,
     1,
     -2
    ],
    [
     2,
     2,
     1,
     1,
     4,
     1,
     1,
     2,
     -1,
     2,
     -1,
     -1,
     1,
     -1,
     1,
     0,
     1,
     0,
     -1
    ],
    [
     0,
     -1,
     1,
     2,
     1,
     4,
     1,
     2,
     0,
     1,
     -1,
     1,
     1,
     0,
     0,
     -1,
     0,
     0,
     -2
    ],
    [
     2,
     2,
     2,
     1,
     1,
     1,
     4,
     1,
     -2,
     1,
     0,
     1,
     1,
     -1,
     1,
     2,
     0,
     1,
     -2
    ],
    [
     2,
     1,
     2,
     2,
     2,
     2,
     1,
     4,
     0,
     1,
     -1,
     -1,
     0,
     -1,
     1,
     -1,
     1,
     2,
     -1
    ],
    [
     -2,
     -2,
     -2,
     1,
     -1,
     0,
     -2,
     0,
     4,
     0,
     1,
     1,
     0,
     1,
     -1,
     -1,
     -2,
     -1,
     1
    ],
    [
     2,
     0,
     0,
     2,
     2,
     1,
     1,
     1,
     0,
     4,
     -1,
     1,
     0,
     -2,
     1,
     0,
     1,
     0,
     -1
    ],
    [
     -2,
     0,
     0,
     0,
     -1,
     -1,
     0,
     -1,
     1,
     -1,
     4,
     0,
     1,
     1,
     0,
     0,
     -1,
     -1,
     0
    ],
    [
     -1,
     -1,
     -1,
     1,
     -1,
     1,
     1,
     -1,
     1,
     1,
     0,
     4,
     1,
     0,
     0,
     1,
     -2,
     -1,
     -1
    ],
    [
     -1,
     1,
     0,
     -1,
     1,
     1,
     1,
     0,
     0,
     0,
     1,
     1,
     4,
     2,
     -1,
     1,
     -1,
     -2,
     0
    ],
    [
     -2,
     0,
     -1,
     -2,
     -1,
     0,
     -1,
     -1,
     1,
     -2,
     1,
     0,
     2,
     4,
     -2,
     1,
     -1,
     -2,
     1
    ],
    [
     2,
     1,
     2,
     2,
     1,
     0,
     1,
     1,
     -1,
     1,
     0,
     0,
     -1,
     -2,
     4,
     0,
     1,
     1,
     -1
    ],
    [
     1,
     2,
     0,
     -1,
     0,
     -1,
     2,
     -1,
     -1,
     0,
     0,
     1,
     1,
     1,
     0,
     4,
     -1,
     0,
     0
    ],
    [
     2,
     1,
     2,
     0,
     1,
     0,
     0,
     1,
     -2,
     1,
     -1,
     -2,
     -1,
     -1,
     1,
     -1,
     4,
     1,
     0
    ],
    [
     2,
     1,
     2,
     1,
     0,
     0,
     1,
     2,
     -1,
     0,
     -1,
     -1,
     -2,
     -2,
     1,
     0,
     1,
     4,
     0
    ],
    [
     -1,
     0,
     -1,
     -2,
     -1,
     -2,
     -2,
     -1,
     1,
     -1,
     0,
     -1,
     0,
     1,
     -1,
     0,
     0,
     0,
     4
    ]
   ],
   "det": 512,
   "even": true,
   "minimum": 4,
   "n_roots": 0,
   "kissing_number": 5200,
   "discriminant_group_invariants": [
    2,
    2,
    128
   ],
   "discriminant_form": {
    "invariants": [
     2,
     2,
     128
    ],
    "q_on_generators_mod2": [
     "1/2",
     "1",
     "25/128"
    ],
    "b_matrix_mod1": [
     [
      "1/2",
      "1/2",
      "0"
     ],
     [
      "1/2",
      "0",
      "0"
     ],
     [
      "0",
      "0",
      "25/128"
     ]
    ]
   },
   "aut_order": 2
  },
  "W512p": {
   "rank": 19,
   "gram": [
    [
     4,
     -2,
     -2,
     -2,
     -1,
     -1,
     -1,
     -1,
     2,
     -2,
     -1,
     1,
     -1,
     -2,
     -2,
     2,
     -1,
     2,
     -1
    ],
    [
     -2,
     4,
     2,
     2,
     2,
     1,
     -1,
     1,
     -1,
     2,
     2,
     1,
     -1,
     0,
     0,
     -2,
     2,
     -2,
     -1
    ],
    [
     -2,
     2,
     4,
     2,
     1,
     2,
     0,
     2,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     -1,
     1,
     -2,
     -1
    ],
    [
     -2,
     2,
     2,
     4,
     0,
     2,
     -1,
     2,
     0,
     1,
     2,
     -1,
     -1,
     1,
     1,
     0,
     1,
     -2,
     0
    ],
    [
     -1,
     2,
     1,
     0,
     4,
     -1,
     -1,
     1,
     0,
     1,
     1,
     2,
     1,
     0,
     -1,
     -1,
     2,
     0,
     -1
    ],
    [
     -1,
     1,
     2,
     2,
     -1,
     4,
     1,
     2,
     -1,
     -1,
     0,
     0,
     0,
     1,
     0,
     0,
     -1,
     -2,
     -1
    ],
    [
     -1,
     -1,
     0,
     -1,
     -1,
     1,
     4,
     -1,
     -2,
     0,
     -2,
     -1,
     1,
     1,
     1,
     -1,
     -2,
     0,
     0
    ],
    [
     -1,
     1,
     2,
     2,
     1,
     2,
     -1,
     4,
     1,
     -1,
     1,
     0,
     0,
     0,
     -1,
     0,
     1,
     -2,
     -1
    ],
    [
     2,
     -1,
     0,
     0,
     0,
     -1,
     -2,
     1,
     4,
     -1,
     0,
     0,
     -1,
     -2,
     -1,
     1,
     0,
     1,
     0
    ],
    [
     -2,
     2,
     0,
     1,
     1,
     -1,
     0,
     -1,
     -1,
     4,
     2,
     0,
     -1,
     1,
     1,
     -2,
     1,
     -1,
     1
    ],
    [
     -1,
     2,
     0,
     2,
     1,
     0,
     -2,
     1,
     0,
     2,
     4,
     1,
     -1,
     0,
     0,
     0,
     2,
     -1,
     0
    ],
    [
     1,
     1,
     0,
     -1,
     2,
     0,
     -1,
     0,
     0,
     0,
     1,
     4,
     0,
     -1,
     -2,
     0,
     1,
     1,
     -1
    ],
    [
     -1,
     -1,
     0,
     -1,
     1,
     0,
     1,
     0,
     -1,
     -1,
     -1,
     0,
     4,
     2,
     1,
     0,
     -1,
     1,
     1
    ],
    [
     -2,
     0,
     0,
     1,
     0,
     1,
     1,
     0,
     -2,
     1,
     0,
     -1,
     2,
     4,
     1,
     0,
     -1,
     -1,
     1
    ],
    [
     -2,
     0,
     1,
     1,
     -1,
     0,
     1,
     -1,
     -1,
     1,
     0,
     -2,
     1,
     1,
     4,
     -1,
     0,
     0,
     2
    ],
    [
     2,
     -2,
     -1,
     0,
     -1,
     0,
     -1,
     0,
     1,
     -2,
     0,
     0,
     0,
     0,
     -1,
     4,
     -1,
     1,
     -1
    ],
    [
     -1,
     2,
     1,
     1,
     2,
     -1,
     -2,
     1,
     0,
     1,
     2,
     1,
     -1,
     -1,
     0,
     -1,
     4,
     -1,
     0
    ],
    [
     2,
     -2,
     -2,
     -2,
     0,
     -2,
     0,
     -2,
     1,
     -1,
     -1,
     1,
     1,
     -1,
     0,
     1,
     -1,
     4,
     1
    ],
    [
     -1,
     -1,
     -1,
     0,
     -1,
     -1,
     0,
     -1,
     0,
     1,
     0,
     -1,
     1,
     1,
     2,
     -1,
     0,
     1,
     4
    ]
   ],
   "det": 512,
   "even": true,
   "minimum": 4,
   "n_roots": 0,
   "kissing_number": 5204,
   "discriminant_group_invariants": [
    2,
    2,
    128
   ],
   "discriminant_form": {
    "invariants": [
     2,
     2,
     128
    ],
    "q_on_generators_mod2": [
     "1",
     "1/2",
     "241/128"
    ],
    "b_matrix_mod1": [
     [
      "0",
      "1/2",
      "1/2"
     ],
     [
      "1/2",
      "1/2",
      "1/2"
     ],
     [
      "1/2",
      "1/2",
      "113/128"
     ]
    ]
   },
   "aut_order": 2
  },
  "W1024": {
   "rank": 19,
   "gram": [
    [
     4,
     -1,
     -1,
     2,
     2,
     2,
     2,
     2,
     2,
     -1,
     -2,
     2,
     2,
     -1,
     -2,
     2,
     -2,
     2,
     2
    ],
    [
     -1,
     4,
     2,
     0,
     -2,
     1,
     1,
     -2,
     -1,
     -1,
     2,
     0,
     -1,
     2,
     2,
     0,
     2,
     -2,
     0
    ],
    [
     -1,
     2,
     4,
     1,
     -1,
     0,
     0,
     -2,
     -2,
     0,
     2,
     1,
     0,
     2,
     0,
     -1,
     1,
     -2,
     1
    ],
    [
     2,
     0,
     1,
     4,
     2,
     2,
     2,
     0,
     1,
     0,
     0,
     1,
     2,
     -1,
     -2,
     1,
     -1,
     1,
     2
    ],
    [
     2,
     -2,
     -1,
     2,
     4,
     0,
     1,
     2,
     1,
     1,
     -1,
     0,
     1,
     -2,
     -2,
     0,
     -2,
     1,
     1
    ],
    [
     2,
     1,
     0,
     2,
     0,
     4,
     1,
     0,
     2,
     -2,
     0,
     2,
     2,
     0,
     0,
     1,
     0,
     1,
     1
    ],
    [
     2,
     1,
     0,
     2,
     1,
     1,
     4,
     1,
     1,
     -1,
     -1,
     0,
     0,
     -1,
     -1,
     2,
     0,
     1,
     1
    ],
    [
     2,
     -2,
     -2,
     0,
     2,
     0,
     1,
     4,
     2,
     -1,
     -2,
     0,
     1,
     -2,
     -1,
     0,
     -2,
     1,
     1
    ],
    [
     2,
     -1,
     -2,
     1,
     1,
     2,
     1,
     2,
     4,
     -2,
     -2,
     1,
     1,
     -1,
     0,
     1,
     -1,
     2,
     1
    ],
    [
     -1,
     -1,
     0,
     0,
     1,
     -2,
     -1,
     -1,
     -2,
     4,
     0,
     -1,
     -1,
     -1,
     -1,
     0,
     -1,
     0,
     -1
    ],
    [
     -2,
     2,
     2,
     0,
     -1,
     0,
     -1,
     -2,
     -2,
     0,
     4,
     0,
     0,
     1,
     1,
     -2,
     2,
     -2,
     -1
    ],
    [
     2,
     0,
     1,
     1,
     0,
     2,
     0,
     0,
     1,
     -1,
     0,
     4,
     2,
     1,
     -1,
     1,
     -1,
     1,
     1
    ],
    [
     2,
     -1,
     0,
     2,
     1,
     2,
     0,
     1,
     1,
     -1,
     0,
     2,
     4,
     -1,
     -2,
     0,
     -1,
     1,
     2
    ],
    [
     -1,
     2,
     2,
     -1,
     -2,
     0,
     -1,
     -2,
     -1,
     -1,
     1,
     1,
     -1,
     4,
     2,
     0,
     1,
     -2,
     0
    ],
    [
     -2,
     2,
     0,
     -2,
     -2,
     0,
     -1,
     -1,
     0,
     -1,
     1,
     -1,
     -2,
     2,
     4,
     -1,
     2,
     -2,
     -1
    ],
    [
     2,
     0,
     -1,
     1,
     0,
     1,
     2,
     0,
     1,
     0,
     -2,
     1,
     0,
     0,
     -1,
     4,
     -1,
     2,
     0
    ],
    [
     -2,
     2,
     1,
     -1,
     -2,
     0,
     0,
     -2,
     -1,
     -1,
     2,
     -1,
     -1,
     1,
     2,
     -1,
     4,
     -1,
     -1
    ],
    [
     2,
     -2,
     -2,
     1,
     1,
     1,
     1,
     1,
     2,
     0,
     -2,
     1,
     1,
     -2,
     -2,
     2,
     -1,
     4,
     0
    ],
    [
     2,
     0,
     1,
     2,
     1,
     1,
     1,
     1,
     1,
     -1,
     -1,
     1,
     2,
     0,
     -1,
     0,
     -1,
     0,
     4
    ]
   ],
   "det": 1024,
   "even": true,
   "minimum": 4,
   "n_roots": 0,
   "kissing_number": 3664,
   "discriminant_group_invariants": [
    2,
    2,
    256
   ],
   "discriminant_form": {
    "invariants": [
     2,
     2,
     256
    ],
    "q_on_generators_mod2": [
     "1/2",
     "1",
     "417/256"
    ],
    "b_matrix_mod1": [
     [
      "1/2",
      "1/2",
      "1/2"
     ],
     [
      "1/2",
      "0",
      "0"
     ],
     [
      "1/2",
      "0",
      "161/256"
     ]
    ]
   },
   "aut_order": 2
  },
  "Lambda19": {
   "rank": 19,
   "gram": [
    [
     4,
     2,
     2,
     -2,
     2,
     -2,
     -2,
     2,
     0,
     0,
     -2,
     -2,
     1,
     0,
     -1,
     -1,
     -1,
     1,
     -2
    ],
    [
     2,
     4,
     2,
     -2,
     2,
     0,
     0,
     2,
     1,
     -1,
     -2,
     -2,
     2,
     -1,
     -2,
     1,
     -2,
     1,
     -1
    ],
    [
     2,
     2,
     4,
     0,
     2,
     0,
     0,
     2,
     -1,
     -1,
     -1,
     -1,
     2,
     -1,
     -2,
     1,
     -1,
     2,
     -2
    ],
    [
     -2,
     -2,
     0,
     4,
     0,
     0,
     0,
     0,
     -1,
     1,
     2,
     2,
     -1,
     1,
     1,
     0,
     1,
     0,
     1
    ],
    [
     2,
     2,
     2,
     0,
     4,
     0,
     0,
     2,
     1,
     -1,
     -1,
     -1,
     1,
     -1,
     -1,
     1,
     0,
     2,
     -1
    ],
    [
     -2,
     0,
     0,
     0,
     0,
     4,
     2,
     -1,
     1,
     -2,
     1,
     0,
     0,
     -1,
     0,
     2,
     1,
     1,
     1
    ],
    [
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     0,
     0,
     0,
     0,
     2,
     4,
     -1,
     1,
     -2,
     1,
     0,
     1,
     -1,
     -1,
     2,
     1,
     0,
     0
    ],
    [
     2,
     2,
     2,
     0,
     2,
     -1,
     -1,
     4,
     1,
     -1,
     -2,
     0,
     0,
     -1,
     0,
     1,
     0,
     2,
     -1
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    [
     0,
     1,
     -1,
     -1,
     1,
     1,
     1,
     1,
     4,
     -2,
     0,
     -1,
     0,
     0,
     1,
     1,
     1,
     1,
     1
    ],
    [
     0,
     -1,
     -1,
     1,
     -1,
     -2,
     -2,
     -1,
     -2,
     4,
     1,
     0,
     -1,
     2,
     0,
     -2,
     -1,
     -2,
     1
    ],
    [
     -2,
     -2,
     -1,
     2,
     -1,
     1,
     1,
     -2,
     0,
     1,
     4,
     0,
     0,
     2,
     0,
     0,
     1,
     -1,
     2
    ],
    [
     -2,
     -2,
     -1,
     2,
     -1,
     0,
     0,
     0,
     -1,
     0,
     0,
     4,
     -1,
     -1,
     1,
     1,
     1,
     0,
     1
    ],
    [
     1,
     2,
     2,
     -1,
     1,
     0,
     1,
     0,
     0,
     -1,
     0,
     -1,
     4,
     -1,
     -2,
     1,
     -2,
     1,
     -1
    ],
    [
     0,
     -1,
     -1,
     1,
     -1,
     -1,
     -1,
     -1,
     0,
     2,
     2,
     -1,
     -1,
     4,
     0,
     -2,
     0,
     -2,
     1
    ],
    [
     -1,
     -2,
     -2,
     1,
     -1,
     0,
     -1,
     0,
     1,
     0,
     0,
     1,
     -2,
     0,
     4,
     -1,
     1,
     0,
     1
    ],
    [
     -1,
     1,
     1,
     0,
     1,
     2,
     2,
     1,
     1,
     -2,
     0,
     1,
     1,
     -2,
     -1,
     4,
     1,
     2,
     0
    ],
    [
     -1,
     -2,
     -1,
     1,
     0,
     1,
     1,
     0,
     1,
     -1,
     1,
     1,
     -2,
     0,
     1,
     1,
     4,
     1,
     0
    ],
    [
     1,
     1,
     2,
     0,
     2,
     1,
     0,
     2,
     1,
     -2,
     -1,
     0,
     1,
     -2,
     0,
     2,
     1,
     4,
     -1
    ],
    [
     -2,
     -1,
     -2,
     1,
     -1,
     1,
     0,
     -1,
     1,
     1,
     2,
     1,
     -1,
     1,
     1,
     0,
     0,
     -1,
     4
    ]
   ],
   "gram_basis": "LLL-reduced basis (all diagonal entries 4); gram_construction_basis is the Gram matrix in the basis of the construction as the orthogonal complement of sqrt(2)D5 in the Leech lattice, and T is the unimodular matrix with T * gram_construction_basis * T^T = gram",
   "gram_construction_basis": [
    [
     4,
     2,
     2,
     2,
     2,
     2,
     -2,
     0,
     0,
     1,
     3,
     0,
     2,
     1,
     -2,
     -4,
     1,
     0,
     2
    ],
    [
     2,
     4,
     2,
     2,
     2,
     2,
     -2,
     0,
     1,
     1,
     3,
     0,
     1,
     2,
     -2,
     -2,
     -1,
     1,
     -1
    ],
    [
     2,
     2,
     4,
     2,
     2,
     2,
     0,
     0,
     -1,
     1,
     3,
     -1,
     1,
     1,
     -1,
     -3,
     2,
     1,
     2
    ],
    [
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     2,
     2,
     4,
     2,
     2,
     0,
     0,
     1,
     0,
     3,
     -1,
     3,
     0,
     -3,
     -1,
     2,
     1,
     2
    ],
    [
     2,
     2,
     2,
     2,
     4,
     2,
     0,
     -1,
     1,
     -2,
     3,
     -2,
     0,
     1,
     -2,
     -2,
     0,
     1,
     1
    ],
    [
     2,
     2,
     2,
     2,
     2,
     4,
     0,
     1,
     1,
     -1,
     2,
     0,
     0,
     2,
     -4,
     -2,
     3,
     3,
     2
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    [
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     0,
     0,
     0,
     0,
     4,
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     -2,
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     0,
     2,
     3,
     1
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    [
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     0,
     0,
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     1,
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     6,
     2,
     1,
     1,
     3,
     4,
     -7,
     2,
     4,
     2,
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     2
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    [
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     1,
     1,
     1,
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     2,
     4,
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     1,
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    [
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     1,
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     1,
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     1,
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     3,
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    [
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     3,
     3,
     3,
     2,
     -2,
     1,
     1,
     1,
     6,
     0,
     4,
     -2,
     0,
     -2,
     1,
     -1,
     2
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    [
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     0,
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     -1,
     -2,
     0,
     -2,
     3,
     1,
     2,
     0,
     4,
     1,
     -2,
     1,
     1,
     0,
     -1,
     0
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    [
     2,
     1,
     1,
     3,
     0,
     0,
     -3,
     4,
     1,
     3,
     4,
     1,
     10,
     -9,
     3,
     3,
     2,
     -5,
     3
    ],
    [
     1,
     2,
     1,
     0,
     1,
     2,
     3,
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     -3,
     0,
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     18,
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     0,
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    [
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     -1,
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     2,
     -1,
     2,
     0,
     1,
     3,
     -9,
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     4,
     -3,
     -7,
     -1
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    [
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     -3,
     -1,
     -2,
     -2,
     0,
     4,
     3,
     -2,
     -2,
     1,
     3,
     -9,
     4,
     10,
     -1,
     -3,
     -1
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    [
     1,
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     2,
     2,
     0,
     3,
     2,
     2,
     -1,
     0,
     1,
     0,
     2,
     0,
     -3,
     -1,
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     3,
     5
    ],
    [
     0,
     1,
     1,
     1,
     1,
     3,
     3,
     -2,
     0,
     -2,
     -1,
     -1,
     -5,
     9,
     -7,
     -3,
     3,
     8,
     0
    ],
    [
     2,
     -1,
     2,
     2,
     1,
     2,
     1,
     2,
     0,
     0,
     2,
     0,
     3,
     -3,
     -1,
     -1,
     5,
     0,
     10
    ]
   ],
   "T": [
    [
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     1,
     0,
     0,
     0,
     -1,
     1,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     -1,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     1,
     0,
     0,
     0
    ],
    [
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     -1,
     0,
     0,
     1,
     0,
     0,
     -1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     1,
     1,
     -1,
     0,
     0,
     0,
     2,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0
    ],
    [
     -1,
     -1,
     1,
     0,
     0,
     1,
     -1,
     -1,
     1,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     0,
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     0,
     0,
     0,
     0,
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     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0
    ],
    [
     0,
     -1,
     0,
     0,
     0,
     0,
     0,
     1,
     1,
     0,
     0,
     -1,
     0,
     1,
     0,
     0,
     0,
     -1,
     0
    ],
    [
     -1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     1,
     1,
     0,
     -1,
     0,
     -1,
     2,
     0,
     0,
     -1,
     0,
     1,
     1,
     0,
     0,
     0,
     0,
     0,
     0
    ],
    [
     1,
     2,
     0,
     -1,
     0,
     -1,
     2,
     0,
     1,
     0,
     0,
     0,
     0,
     0,
     0,
     0,
     1,
     -1,
     0
    ],
    [
     -9,
     -7,
     1,
     3,
     2,
     2,
     -10,
     0,
     0,
     1,
     0,
     -2,
     0,
     0,
     -1,
     -2,
     -1,
     2,
     1
    ]
   ],
   "det_T": -1,
   "det": 128,
   "even": true,
   "minimum": 4,
   "n_roots": 0,
   "kissing_number": 10668,
   "discriminant_group_invariants": [
    2,
    2,
    2,
    2,
    8
   ],
   "discriminant_form": {
    "invariants": [
     2,
     2,
     2,
     2,
     8
    ],
    "q_on_generators_mod2": [
     "1",
     "1",
     "0",
     "0",
     "3/8"
    ],
    "b_matrix_mod1": [
     [
      "0",
      "1/2",
      "0",
      "0",
      "0"
     ],
     [
      "1/2",
      "0",
      "1/2",
      "0",
      "0"
     ],
     [
      "0",
      "1/2",
      "0",
      "1/2",
      "0"
     ],
     [
      "0",
      "0",
      "1/2",
      "0",
      "0"
     ],
     [
      "0",
      "0",
      "0",
      "0",
      "3/8"
     ]
    ]
   },
   "aut_order": 23592960
  }
 },
 "gluings": {
  "W512": {
   "overlattice_gram": [
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    [
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    [
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    [
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    [
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     0,
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    [
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    [
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