Tag 02AE

\(\begin{pmatrix} 4 & 1 \\ 1 & 4 \end{pmatrix} \oplus \langle 20 \rangle\)

(4 1; 1 4) + <20>·invariant lattice of the symplectic K3 group G_70 of Hashimoto, class a

Invariants

Rank3
Signature \((n_+, n_-)\)(3, 0)
Determinant\(300\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, e_{2}, e_{3}\).

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

\(e_{1}\)\(e_{2}\)\(e_{3}\)
\(e_{1}\)410
\(e_{2}\)140
\(e_{3}\)0020
 
[[4, 1, 0], [1, 4, 0], [0, 0, 20]]

Integral lattice

Parityeven
Discriminant group\(\mathbb{Z}/5 \oplus \mathbb{Z}/60\)
Unimodularno
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 \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \zeta^\Sigma(s - 2)\, L^\Sigma(s - 1, \chi_{-300n})\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number4

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 4q^{4} + 2q^{6} + 2q^{10} + O(q^{13}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots6
\(R(L) = \mathbb{Z}\Phi(L)\)\(\cong \langle 20 \rangle \oplus \langle 6 \rangle \oplus \langle 10 \rangle\)
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 \(\Phi(L)\)

ComponentTypeValues of \(b(r, r)\)Number of roots\(\mathbb{Z}\Phi_c\)
\(\Phi_{1}\)\(A_{1}\)\(\{20\}\)2\(\cong \langle 20 \rangle\)
\(\Phi_{2}\)\(A_{1}\)\(\{6\}\)2\(\cong \langle 6 \rangle\)
\(\Phi_{3}\)\(A_{1}\)\(\{10\}\)2\(\cong \langle 10 \rangle\)

Simple roots

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

Simple rootCoordinates\(b(r, r)\)
\(\alpha^{(1)}_{1}\)\(e_{3}\)\(20\)
\(\alpha^{(2)}_{1}\)\(e_{1} - e_{2}\)\(6\)
\(\alpha^{(3)}_{1}\)\(e_{1} + e_{2}\)\(10\)

Notes

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. Row 70 of Table 10.3 of Hashimoto gives the invariant lattice \(\Lambda^G\) for the group \(\mathfrak{G}_{70} \cong \mathfrak{S}_5\) of Table 10.2, and this record is \(\Lambda^G\). The genus of \(\Lambda^G\) in row 70 has two classes, and the table lists both; the other is the record (4 2 2; 2 6 1; 2 1 16).

  • 02AX \(\Lambda_{\mathfrak{G}_{70}}(-1)\) A lattice \(M\) with \(M(-1) \cong L^\perp\) in \(\mathrm{II}_{3,19}\), where \(L = \Lambda^G\) for the group \(\mathfrak{G}_{70}\) of Table 10.2 of Hashimoto.
  • 027E \(U^{3} \oplus E_{8}(-1)^{2}\) The K3 lattice \(\mathrm{II}_{3,19}\), in which \(L\) is the invariant lattice \(\Lambda^G\) of the group \(\mathfrak{G}_{70}\) 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.