\(\Lambda_{\mathfrak{G}_{56}}(-1)\)

Let \(\Lambda = \mathrm{II}_{3,19}\) be the K3 lattice and \(G \subset O(\Lambda)\) a finite group of symplectic isometries. The coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\) is negative definite of rank 18 and has no vector of norm \(-2\); this record is \(\Lambda_G(-1)\) for the groups \(\mathfrak{G}_{56} \cong \Gamma_{25} a_1\), \(\mathfrak{G}_{41} \cong \Gamma_7 a_1\) of Table 10.2 of Hashimoto, whose invariant lattice \(\Lambda^G\) is in row 56 of Table 10.3.

The Gram tensor is the LLL reduction of the coinvariant lattice lattices[6,11] of Höhn and Mason: the orthogonal complement in the Leech lattice of the fixed-point lattice \(B\) of rank 6 that is number 11 of rank 6 in their Table 1. It satisfies their K3 criterion \(\operatorname{rk} B \geq 5\), \(\operatorname{rk} B - \operatorname{rk} A_B \geq 2\), and among the coinvariant lattices that satisfy it, it is the only one whose rank and discriminant form are those of \(\Lambda_G(-1)\): its discriminant form is the form \(q\) of row 56 of Table 10.3. The sum of its twist by \(-1\) and \(\Lambda^G\) therefore glues to an even unimodular lattice of signature \((3, 19)\), on which the stabilizer of \(B\) in \(\mathrm{Co}_0\) acts with this coinvariant lattice.