Tag 0251

\(E_{8}^{2} \oplus A_{3}\)

E8^2 + A3

Invariants

Rank19
Signature \((n_+, n_-)\)(19, 0)
Determinant\(4\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{19}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}, \dots, e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}, \dots, e_{16}\}\) \(\oplus\) \(\mathbb{Z}\{e_{17}, e_{18}, e_{19}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)
\(e_{1}\)20-10000000000000000
\(e_{2}\)020-1000000000000000
\(e_{3}\)-102-1000000000000000
\(e_{4}\)0-1-12-100000000000000
\(e_{5}\)000-12-10000000000000
\(e_{6}\)0000-12-1000000000000
\(e_{7}\)00000-12-100000000000
\(e_{8}\)000000-1200000000000
\(e_{9}\)0000000020-100000000
\(e_{10}\)00000000020-10000000
\(e_{11}\)00000000-102-10000000
\(e_{12}\)000000000-1-12-1000000
\(e_{13}\)00000000000-12-100000
\(e_{14}\)000000000000-12-10000
\(e_{15}\)0000000000000-12-1000
\(e_{16}\)00000000000000-12000
\(e_{17}\)00000000000000002-10
\(e_{18}\)0000000000000000-12-1
\(e_{19}\)00000000000000000-12
 
[[2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [-1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 2, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, -1, 2, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2, -1], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 2]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/4\)
Unimodularno
Integral overlattices \(L \subseteq M \subseteq L^*\)2
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 18)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 18)\, L^\Sigma(s - 9, \chi_{-4n})\)
Genus symbol\(\mathrm{II}_{19,0}\ \bigl(2\colon 1^{-18}:[4^{-1}]_{3}\bigr)\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(2\)
Kissing number492
Order of the isometry group\(46601485009551360000 = 2^{33} \cdot 3^{11} \cdot 5^{4} \cdot 7^{2}\)

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 492q^{2} + 67686q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{2\}\)
Number of roots498
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong E_{8}^{2} \oplus A_{3}\)
Rank of \(R(L)\)19
\([L : R(L)]\)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\(\cong E_{8}\)
\(\Phi_{2}\)\(E_{8}\)\(\{2\}\)240\(\cong E_{8}\)
\(\Phi_{3}\)\(C_{3}\)\(\{2, 4\}\)18\(\cong A_{3}\)

Simple roots

\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{9}\)\(2\)
\(\alpha^{(1)}_{2}\)\(e_{10}\)\(2\)
\(\alpha^{(1)}_{3}\)\(e_{11}\)\(2\)
\(\alpha^{(1)}_{4}\)\(e_{12}\)\(2\)
\(\alpha^{(1)}_{5}\)\(e_{13}\)\(2\)
\(\alpha^{(1)}_{6}\)\(e_{14}\)\(2\)
\(\alpha^{(1)}_{7}\)\(e_{15}\)\(2\)
\(\alpha^{(1)}_{8}\)\(e_{16}\)\(2\)
\(\alpha^{(2)}_{1}\)\(e_{1}\)\(2\)
\(\alpha^{(2)}_{2}\)\(e_{2}\)\(2\)
\(\alpha^{(2)}_{3}\)\(e_{3}\)\(2\)
\(\alpha^{(2)}_{4}\)\(e_{4}\)\(2\)
\(\alpha^{(2)}_{5}\)\(e_{5}\)\(2\)
\(\alpha^{(2)}_{6}\)\(e_{6}\)\(2\)
\(\alpha^{(2)}_{7}\)\(e_{7}\)\(2\)
\(\alpha^{(2)}_{8}\)\(e_{8}\)\(2\)
\(\alpha^{(3)}_{1}\)\(e_{18}\)\(2\)
\(\alpha^{(3)}_{2}\)\(e_{19}\)\(2\)
\(\alpha^{(3)}_{3}\)\(e_{17} - e_{19}\)\(4\)

Morphisms