Tag 02AN
\(\begin{pmatrix} 6 & 0 & 3 \\ 0 & 6 & 3 \\ 3 & 3 & 8 \end{pmatrix}\)
(6 0 3; 0 6 3; 3 3 8)·invariant lattice of the symplectic K3 group G_79 of Hashimoto, class b
Invariants
| Rank | 3 |
|---|---|
| Signature \((n_+, n_-)\) | (3, 0) |
| Determinant | \(180\) |
Gram tensor
The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).
| \(e_{1}\) | \(e_{2}\) | \(e_{3}\) | |
|---|---|---|---|
| \(e_{1}\) | 6 | 0 | 3 |
| \(e_{2}\) | 0 | 6 | 3 |
| \(e_{3}\) | 3 | 3 | 8 |
[[6, 0, 3], [0, 6, 3], [3, 3, 8]]
Integral lattice
| Parity | even |
|---|---|
| Discriminant group | \(\mathbb{Z}/3 \oplus \mathbb{Z}/60\) |
| Unimodular | no |
| Integral overlattices \(L \subseteq M \subseteq L^*\) | 2 |
| Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\) | \(\{2, 3, 5\}\) |
| Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\) | \(\zeta^\Sigma(X_0, s) = \zeta^\Sigma(s - 2)\) |
| Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\) | \(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-180n})\) |
Definite lattice
| Minimum of \(b(x, x)\), \(x \neq 0\) | \(6\) |
|---|---|
| Kissing number | 4 |
Theta series
\[ \sum_{x \in L} q^{b(x, x)} = 1 + 4q^{6} + 8q^{8} + 4q^{12} + O(q^{13}) \]Roots
| Root lattice: \(L = \mathbb{Z}\Phi(L)\) | no |
|---|---|
| Number of roots | 10 |
| \(R(L) = \mathbb{Z}\Phi(L)\) | \(\cong \mathrm{I}_{2,0}(6) \oplus \langle 20 \rangle\) |
| Rank of \(R(L)\) | 3 |
| \([L : R(L)]\) | 2 |
| \(R(L)\) is primitive in \(L\) | no: \(L / R(L) \cong \mathbb{Z}/2\) |
Components of \(\Phi(L)\)
| Component | Type | Values of \(b(r, r)\) | Number of roots | \(\mathbb{Z}\Phi_c\) |
|---|---|---|---|---|
| \(\Phi_{1}\) | \(B_{2}\) | \(\{6, 12\}\) | 8 | \(\cong \mathrm{I}_{2,0}(6)\) |
| \(\Phi_{2}\) | \(A_{1}\) | \(\{20\}\) | 2 | \(\cong \langle 20 \rangle\) |
Simple roots
\(\alpha^{(c)}_j\) is simple root \(j\) of the component \(\Phi_c\).
| Simple root | Coordinates | \(b(r, r)\) |
|---|---|---|
| \(\alpha^{(1)}_{1}\) | \(e_{1} - e_{2}\) | \(12\) |
| \(\alpha^{(1)}_{2}\) | \(e_{2}\) | \(6\) |
| \(\alpha^{(2)}_{1}\) | \(e_{1} + e_{2} - 2e_{3}\) | \(20\) |
Notes
Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 79 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{79} \cong \mathfrak{A}_6\) of Table 10.2, and this record is \(\Lambda^G\). The genus of \(\Lambda^G\) in row 79 has two classes, and the table lists both; the other is the record (2 1; 1 8) + <12>.
Related lattices
Morphisms
References
- K. Hashimoto, "Finite symplectic actions on the K3 lattice", arXiv:1012.2682, Tables 10.2 and 10.3.