The E₆ genus of the shorter Conway–Wales lattice

We classify the definite integral structures of type \({}^2E_6\) over the Gaussian integers which are everywhere locally isomorphic to the trace-parity shortening of the standard integral Albert algebra. The genus has exactly eight classes, two at each of the heights \(1\), \(2\), \(4\) and \(5\), and we determine the orders of their automorphism groups. The two classes of height five are the shorter Conway–Wales lattice and its complex conjugate, equipped with their cubic forms. The classification is proved by a Venkov-type argument whose modular input is a height bound obtained from the exceptional theta lift for \(E_6\times\mathrm{SU}(2,1)\), and it is confirmed independently by the mass formula and by an enumeration of dyadic neighbours. The exceptional lift has five-dimensional image, and its cuspidal part spans the two-dimensional cuspidal cohomology \(H^2_{\mathrm{cusp}}(\Gamma_{\mathrm{Pic}},V_{(3,3)})\) of the Picard modular group over \(\mathbb Z[i]\). We also describe the eight classes as orbits of integral markings of the rank-\(56\) Freudenthal module of \(E_7\). The archive e6_certificate.zip contains the computational certificate for Section 6 of the paper.