Tag 0193

\(\Lambda_{15}\)

Lambda15·LAMBDA15

Invariants

Rank15
Signature \((n_+, n_-)\)(15, 0)
Determinant\(512\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{15}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)
\(e_{1}\)4-20000000000011
\(e_{2}\)-24-2200000000-100
\(e_{3}\)0-24002000000211
\(e_{4}\)020422000000011
\(e_{5}\)000242002100000
\(e_{6}\)002224221200112
\(e_{7}\)0000024202000-11
\(e_{8}\)000002240200102
\(e_{9}\)000021004200000
\(e_{10}\)000012222422112
\(e_{11}\)000000000242211
\(e_{12}\)000000000224122
\(e_{13}\)0-12001010121402
\(e_{14}\)101101-100112042
\(e_{15}\)101102120212224
 
[[4, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1], [-2, 4, -2, 2, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, -2, 4, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 1, 1], [0, 2, 0, 4, 2, 2, 0, 0, 0, 0, 0, 0, 0, 1, 1], [0, 0, 0, 2, 4, 2, 0, 0, 2, 1, 0, 0, 0, 0, 0], [0, 0, 2, 2, 2, 4, 2, 2, 1, 2, 0, 0, 1, 1, 2], [0, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 0, 0, -1, 1], [0, 0, 0, 0, 0, 2, 2, 4, 0, 2, 0, 0, 1, 0, 2], [0, 0, 0, 0, 2, 1, 0, 0, 4, 2, 0, 0, 0, 0, 0], [0, 0, 0, 0, 1, 2, 2, 2, 2, 4, 2, 2, 1, 1, 2], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 2, 2, 1, 1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 2, 4, 1, 2, 2], [0, -1, 2, 0, 0, 1, 0, 1, 0, 1, 2, 1, 4, 0, 2], [1, 0, 1, 1, 0, 1, -1, 0, 0, 1, 1, 2, 0, 4, 2], [1, 0, 1, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 2, 4]]

Integral lattice

Parityeven
Discriminant group\((\mathbb{Z}/2)^{6} \oplus \mathbb{Z}/8\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2981
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 14)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 14)\, L^\Sigma(s - 7, \chi_{-512n})\)
Genus symbol\(\mathrm{II}_{15,0}\ \bigl(2\colon 1^{8}:2^{6}:[8^{1}]_{7}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number2340
Order of the isometry group\(41287680 = 2^{17} \cdot 3^{2} \cdot 5 \cdot 7\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 2340q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots0
\(R(L) = \mathbb{Z}\Phi(L)\)\(0\)
Rank of \(R(L)\)0
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Notes

The Gram tensor is the section GRAM of the entry LAMBDA15 of the Catalogue of Lattices.

The catalogue entry gives this reference text: 1. SPLAG p. 157. 2. W.Plesken, M.Pohst, Constructing integral lattices with prescribed minimum II, Math. Comp. Vol 60, N. 202 (1993) 817-825

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. For the group \(\mathfrak{G}_{21} \cong C_2^4\) of symplectic isometries in Table 10.2 of Hashimoto, the coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) twisted by \(-1\) is this lattice; \(\Lambda^G\) is in row 21 of Table 10.3. The coinvariant lattice lattices[9,1] of Höhn and Mason, the orthogonal complement in the Leech lattice of their fixed-point lattice number 1 of rank 9, is isometric to this lattice, and its discriminant form is the form \(q\) of row 21 of Table 10.3.

  • 029V \(U^{3} \oplus \langle -4 \rangle\) A lattice \(M\) with \(M(2) \cong L(-1)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-1) = \Lambda_G\) for the group \(\mathfrak{G}_{21}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(-1)\) is the coinvariant lattice \(\Lambda_G\) of the group \(\mathfrak{G}_{21}\) of Table 10.2 of Hashimoto.
  • 028S \(\Lambda_{24}\) The Leech lattice \(\Lambda\), in which \(L\) is the orthogonal complement of the sublattice fixed by a subgroup of \(\mathrm{Co}_0\).

Morphisms

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry LAMBDA15. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/LAMBDA15.html
  2. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  3. G. Höhn and G. Mason, "The 290 fixed-point sublattices of the Leech lattice", arXiv:1505.06420, Table 1 and its ancillary Magma file lattices.txt, entry lattices[9,1].