Tag 02BR

\(\mathrm{BGF.2.213}\)

BGF.2.213·BGF.2.213

Invariants

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}\).

[[2, 1], [1, 3]]

Parity	odd
Discriminant group	\(\mathbb{Z}/5\)
Unimodular	no
Integral overlattices \(L \subseteq M \subseteq L^*\)	1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)	\(\{2, 5\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{1} \det L\)	\(\chi_{-20} = \left(\frac{-20}{\cdot}\right)\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)	\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 1)\, L^\Sigma(s - 1, \chi_{-20})}{L^\Sigma(s, \chi_{-20})}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)	\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 1)}{L^\Sigma(s, \chi_{-20})}\)

Minimum of \(b(x, x)\), \(x \neq 0\)	\(2\)
Kissing number	2

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 2q^{2} + 4q^{3} + 4q^{7} + 2q^{8} + 2q^{10} + 4q^{12} + O(q^{13}) \]

Root lattice: \(L = \mathbb{Z}\Phi(L)\)	no
Number of roots	4
\(R(L) = \mathbb{Z}\Phi(L)\)	\(\cong \langle 10 \rangle \oplus \langle 2 \rangle\)
Rank of \(R(L)\)	2
\([L : R(L)]\)	2
\(R(L)\) is primitive in \(L\)	no: \(L / R(L) \cong \mathbb{Z}/2\)

Component	Type	Values of \(b(r, r)\)	Number of roots	\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)	\(A_{1}\)	\(\{10\}\)	2	\(\cong \langle 10 \rangle\)
\(\Phi_{2}\)	\(A_{1}\)	\(\{2\}\)	2	\(\cong \langle 2 \rangle\)

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

Simple root	Coordinates	\(b(r, r)\)
\(\alpha^{(1)}_{1}\)	\(e_{1} - 2e_{2}\)	\(10\)
\(\alpha^{(2)}_{1}\)	\(e_{1}\)	\(2\)

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry BGF.2.213 of the Catalogue of Lattices.

G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry BGF.2.213. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/union.gz