Tag 029R
\(\begin{pmatrix} -3 & -1 & -1 \\ -1 & -3 & 1 \\ -1 & 1 & -3 \end{pmatrix} \oplus \langle 1 \rangle^{2}\)
(-3 -1 -1; -1 -3 1; -1 1 -3) + <1>^2·invariant lattice of the symplectic K3 group G_12 of Hashimoto
Invariants
| Rank | 5 |
|---|---|
| Signature \((n_+, n_-)\) | (2, 3) |
| Determinant | \(-16\) |
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, \(L = {}\)\(\mathbb{Z}\{e_{1}, e_{2}, e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\), an orthogonal sum.
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | \(e_{4}\) | \(e_{5}\) | |
|---|---|---|---|---|---|
| \(e_{1}\) | -3 | -1 | -1 | 0 | 0 |
| \(e_{2}\) | -1 | -3 | 1 | 0 | 0 |
| \(e_{3}\) | -1 | 1 | -3 | 0 | 0 |
| \(e_{4}\) | 0 | 0 | 0 | 1 | 0 |
| \(e_{5}\) | 0 | 0 | 0 | 0 | 1 |
[[-3, -1, -1, 0, 0], [-1, -3, 1, 0, 0], [-1, 1, -3, 0, 0], [0, 0, 0, 1, 0], [0, 0, 0, 0, 1]]
Integral lattice
| Parity | odd |
|---|---|
| Discriminant group | \((\mathbb{Z}/4)^{2}\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 5 |
| 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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 4)\, L^\Sigma(s - 2, \chi_{-16n})\) |
Indefinite lattice
| Isotropic | yes |
|---|
Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | yes |
|---|---|
| \(L = \mathbb{Z}\Phi_S(L)\) for | \(S = \{-8, -2, 1\}\) |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(L\) |
| Rank of \(R(L)\) | 5 |
| \([L : R(L)]\) | 1 |
| \(R(L)\) is primitive in \(L\) | yes |
| Proof | Each \(r_i\) 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 \(L\).
| Root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(r_{1}\) | \(e_{4}\) | \(1\) |
| \(r_{2}\) | \(e_{5}\) | \(1\) |
| \(r_{3}\) | \(e_{1} + e_{2}\) | \(-8\) |
| \(r_{4}\) | \(e_{1} + e_{3}\) | \(-8\) |
| \(r_{5}\) | \(e_{1} + e_{4}\) | \(-2\) |
Notes
Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 12 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{12} \cong Q_8\) of Table 10.2, and this record is the lattice \(M\) with \(\Lambda^G = M(-2)\); the corpus records \(M\) and not its twist.
Related lattices
Morphisms
References
- K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.