Tag 0059

\(\mathrm{I}_{2,3}\)

I_{2,3}

Invariants

Rank5
Signature \((n_+, n_-)\)(2, 3)
Determinant\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{5}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)
\(e_{1}\)10000
\(e_{2}\)01000
\(e_{3}\)00-100
\(e_{4}\)000-10
\(e_{5}\)0000-1
 
[[1, 0, 0, 0, 0], [0, 1, 0, 0, 0], [0, 0, -1, 0, 0], [0, 0, 0, -1, 0], [0, 0, 0, 0, -1]]

Integral lattice

Parityodd
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 4)\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 4)\, L^\Sigma(s - 2, \chi_{-n})\)
Genus symbol\(\mathrm{I}_{2,3}\)
Class number of the genus1

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-1, 1\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)5
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\).

Roots that generate \(\mathbb{Z}\Phi(L)\)

The \(r_i\) are a basis of \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{1}\)\(1\)
\(r_{2}\)\(e_{2}\)\(1\)
\(r_{3}\)\(e_{3}\)\(-1\)
\(r_{4}\)\(e_{4}\)\(-1\)
\(r_{5}\)\(e_{5}\)\(-1\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice. Row 40 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G = \langle 4 \rangle^3 \oplus \langle -4 \rangle^2\), which is this lattice twisted by \(-4\), for the groups \(\mathfrak{G}_{40} \cong Q_8 * Q_8\), \(\mathfrak{G}_{13} \cong Q_8\) and \(\mathfrak{G}_{24} \cong Q_8 * C_4\) of Table 10.2.

  • 02BE \(\Lambda_{\mathfrak{G}_{40}}(-1)\) A lattice \(M\) with \(M(-1) \cong L(-4)^\perp\) in \(\mathrm{II}_{3,19}\), where \(L(-4) = \Lambda^G\) for the group \(\mathfrak{G}_{40}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L(-4)\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{40}\) of Table 10.2 of Hashimoto.

Morphisms

References

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