Tag 0094

\(E_{8}\)

E8

Invariants

Rank8
Signature \((n_+, n_-)\)(8, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{8}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)
\(e_{1}\)20-100000
\(e_{2}\)020-10000
\(e_{3}\)-102-10000
\(e_{4}\)0-1-12-1000
\(e_{5}\)000-12-100
\(e_{6}\)0000-12-10
\(e_{7}\)00000-12-1
\(e_{8}\)000000-12
 
[[2, 0, -1, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0], [0, -1, -1, 2, -1, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0], [0, 0, 0, 0, 0, -1, 2, -1], [0, 0, 0, 0, 0, 0, -1, 2]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((8, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{4} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 7)\, \zeta^\Sigma(s - 4)}{\zeta^\Sigma(s - 3)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 7)}{\zeta^\Sigma(s - 3)}\)
Genus symbol\(\mathrm{II}_{8,0}\)
Class number of the genus1

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number240
Order of the isometry group\(696729600 = 2^{14} \cdot 3^{5} \cdot 5^{2} \cdot 7\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 240q^{2} + 2160q^{4} + 6720q^{6} + 17520q^{8} + O(q^{9}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots240
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong E_{8}\)
Rank of \(R(L)\)8
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes

Components of \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)

Simple roots

Simple rootCoordinates\(b(r, r)\)
\(\alpha_{1}\)\(e_{1}\)\(2\)
\(\alpha_{2}\)\(e_{2}\)\(2\)
\(\alpha_{3}\)\(e_{3}\)\(2\)
\(\alpha_{4}\)\(e_{4}\)\(2\)
\(\alpha_{5}\)\(e_{5}\)\(2\)
\(\alpha_{6}\)\(e_{6}\)\(2\)
\(\alpha_{7}\)\(e_{7}\)\(2\)
\(\alpha_{8}\)\(e_{8}\)\(2\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. For the group \(\mathfrak{G}_{1} \cong C_2\) of symplectic isometries in Table 10.2 of Hashimoto, the coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) twisted by \(-1\) is this lattice twisted by 2; \(\Lambda^G\) is in row 1 of Table 10.3. The coinvariant lattice lattices[16,1] of Höhn and Mason, the orthogonal complement in the Leech lattice of their fixed-point lattice number 1 of rank 16, is isometric to this lattice twisted by 2, and its discriminant form is the form \(q\) of row 1 of Table 10.3.

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

Morphisms

References

  1. K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.
  2. 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[16,1].