{
 "description": "The seven root-free admissible rank-19 lattices with 160 <= det <= 224 of Section 3.3 / SM Section S1.2 (P160 ... P224, one per determinant), their exact invariants, and integer isometries U with U^T * B * U = A (A, B = Gram matrices named by A/A_ref and B_ref; det U = +-1) to reference Gram matrices from the literature: P160 = Lambda_M20, P180 = Lambda_A6, P196 = Lambda_L2(7) (the Leech-lattice models of these K3 coinvariant lattices distributed by Hoehn-Mason), P160/P180/P196/P208/P216/P220 = lattices or sections in Martinet's tables, P160 and P208 = Nebe-Sloane catalogue entries attributed to Kallus. rank_fixed_by_O_0 is the rank of the sublattice fixed pointwise by O_0 = ker(Aut(L) -> O(L^*/L)); it is 0 for a coinvariant lattice, so P208, P216, P220, P224 are not coinvariant lattices. P216 is not isometric to Lambda_M9 (HM no. 84; different discriminant groups) and no reference Gram matrix of determinant 224 was found. Reference Gram matrices are reproduced under 'references' with their sources for the purpose of the isometry check.",
 "lattices": {
  "P160": {
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     -1
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    [
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     2,
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     1,
     3,
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   ],
   "det": 160,
   "minimum": 4,
   "kissing_number": 9480,
   "n_roots": 0,
   "smith_invariants": [
    40,
    2,
    2
   ],
   "theta_N2_N4_N6_N8": [
    0,
    9480,
    285600,
    3339420
   ],
   "aut_order": 92160,
   "image_in_O_of_discriminant_group": 96,
   "O_0_order": 960,
   "rank_fixed_by_O_0": 0
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    [
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    [
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   ],
   "det": 180,
   "minimum": 4,
   "kissing_number": 8910,
   "n_roots": 0,
   "smith_invariants": [
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    3
   ],
   "theta_N2_N4_N6_N8": [
    0,
    8910,
    269800,
    3142980
   ],
   "aut_order": 11520,
   "image_in_O_of_discriminant_group": 32,
   "O_0_order": 360,
   "rank_fixed_by_O_0": 0
  },
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    [
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    [
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    [
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     0,
     0,
     -2,
     4,
     -3,
     -3,
     -2,
     -1,
     2,
     1,
     -3,
     -2,
     6,
     5
    ],
    [
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     2,
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     5,
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     -4,
     4,
     -1,
     -1,
     -4,
     3,
     -3,
     -3,
     -5,
     -3,
     5,
     12
    ]
   ],
   "det": 196,
   "minimum": 4,
   "kissing_number": 8526,
   "n_roots": 0,
   "smith_invariants": [
    28,
    7
   ],
   "theta_N2_N4_N6_N8": [
    0,
    8526,
    258720,
    3010980
   ],
   "aut_order": 5376,
   "image_in_O_of_discriminant_group": 32,
   "O_0_order": 168,
   "rank_fixed_by_O_0": 0
  },
  "P208": {
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    [
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    [
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 "references": {
  "HM_no_65": {
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   "source": "Hoehn and Mason, 'The 290 fixed-point sublattices of the Leech lattice', J. Algebra 448 (2016) 618, arXiv:1505.06420, ancillary file anc/lattices.txt (rank-19 coinvariant lattices; Table 1 row number = 64 + j)"
  },
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     1,
     0
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    [
     -2,
     1,
     -1,
     2,
     2,
     2,
     1,
     -2,
     -1,
     1,
     0,
     -1,
     -1,
     -2,
     -1,
     1,
     1,
     4,
     0
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    [
     -2,
     1,
     -2,
     1,
     1,
     0,
     1,
     -1,
     -1,
     -1,
     0,
     0,
     0,
     0,
     1,
     -1,
     0,
     0,
     4
    ]
   ],
   "source": "Nebe-Sloane Catalogue of Lattices, entries dim19kis9480min4det160, dim19kis8634min4det192, dim19kis8292min4det208 ('Perfect lattice found by Yoav Kallus')"
  },
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     0,
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     0,
     1,
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     -1,
     1,
     -1,
     -1,
     0,
     0,
     0,
     0,
     4,
     1,
     2,
     0
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    [
     -2,
     0,
     -2,
     2,
     1,
     0,
     -1,
     -2,
     2,
     0,
     -1,
     -1,
     2,
     2,
     -1,
     1,
     4,
     0,
     -1
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    [
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     2,
     0,
     0,
     0,
     0,
     1,
     1,
     -1,
     -2,
     0,
     -1,
     -1,
     1,
     2,
     0,
     4,
     -1
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    [
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     0,
     -1,
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     0,
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     1,
     1,
     1,
     -2,
     -1,
     2,
     0,
     -1,
     -1,
     4
    ]
   ],
   "source": "Hoehn and Mason, 'The 290 fixed-point sublattices of the Leech lattice', J. Algebra 448 (2016) 618, arXiv:1505.06420, ancillary file anc/lattices.txt (rank-19 coinvariant lattices; Table 1 row number = 64 + j)"
  },
  "Martinet_LL19c": {
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     0,
     0,
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    [
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     1,
     1,
     1,
     2,
     1,
     0,
     1,
     2,
     1,
     1,
     1,
     0,
     0,
     4,
     0,
     -1,
     1
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     -1,
     0,
     0,
     0,
     0,
     2,
     0,
     0,
     1,
     -1,
     1,
     0,
     0,
     0,
     4,
     0,
     1
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    [
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     2,
     1,
     1,
     0,
     -2,
     2,
     1,
     0,
     0,
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     0,
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     1,
     1,
     -1,
     1,
     0,
     0,
     1,
     1,
     -1,
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    ]
   ],
   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
  },
  "Nebe_Sloane_det192_Kallus": {
   "name": "Nebe-Sloane dim19kis8634min4det192 (Kallus)",
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     2,
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     1,
     -1,
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     2,
     -1,
     -1,
     4,
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    [
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     -1,
     0,
     0,
     -1,
     2,
     -1,
     -1,
     1,
     1,
     0,
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     -1,
     1,
     -2,
     4,
     -1,
     -2
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    [
     -1,
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     2,
     -1,
     -1,
     0,
     0,
     0,
     -1,
     1,
     1,
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     -1,
     -1,
     1,
     -1,
     4,
     1
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    [
     -1,
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     1,
     0,
     0,
     0,
     -1,
     1,
     0,
     0,
     -1,
     0,
     1,
     1,
     0,
     1,
     -2,
     1,
     4
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   ],
   "source": "Nebe-Sloane Catalogue of Lattices, entries dim19kis9480min4det160, dim19kis8634min4det192, dim19kis8292min4det208 ('Perfect lattice found by Yoav Kallus')"
  },
  "HM_no_84": {
   "name": "HM no. 84 = Lambda_M9 (row j=20)",
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     2,
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     1,
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     2,
     3,
     2,
     1,
     2,
     2,
     2,
     2,
     1,
     2,
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     2,
     2
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     0,
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     0,
     2,
     2,
     1,
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
  },
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     1,
     4,
     1,
     0
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    [
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     -1,
     2,
     2,
     2,
     1,
     -2,
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     0,
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     1,
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     0,
     0,
     0,
     0,
     1,
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     0,
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   "source": "Nebe-Sloane Catalogue of Lattices, entries dim19kis9480min4det160, dim19kis8634min4det192, dim19kis8292min4det208 ('Perfect lattice found by Yoav Kallus')"
  },
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     0,
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     1,
     1,
     2,
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     0,
     0,
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     1,
     0,
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     0,
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     0,
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     0,
     4,
     0,
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     1,
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     0,
     0,
     0,
     1,
     0,
     1,
     0,
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     0,
     4
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   ],
   "source": "Hoehn and Mason, 'The 290 fixed-point sublattices of the Leech lattice', J. Algebra 448 (2016) 618, arXiv:1505.06420, ancillary file anc/lattices.txt (rank-19 coinvariant lattices; Table 1 row number = 64 + j)"
  },
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     0,
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     0,
     0,
     1,
     -1,
     1,
     -1,
     -1,
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
  },
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
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     0,
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     1,
     0,
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     0,
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     0,
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     1,
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     1,
     0,
     0,
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     1,
     0,
     0,
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     0,
     -2,
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     1,
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     0,
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     1,
     1,
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   "source": "Nebe-Sloane Catalogue of Lattices, entries dim19kis9480min4det160, dim19kis8634min4det192, dim19kis8292min4det208 ('Perfect lattice found by Yoav Kallus')"
  },
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
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   "source": "J. Martinet, online tables of lattices, files Lambda.gp and AntiLaminO23.gp (https://jamartin.perso.math.cnrs.fr/Lattices/)"
  }
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