## Lattices obtained from the extremal unimodular 72 dimensional lattice

An entry from the Catalogue of Lattices, which is a joint project of

Gabriele Nebe, RWTH Aachen University
(nebe@math.rwth-aachen.de)
and

Neil J. A. Sloane, AT&T Shannon Labs
(njasloane@gmail.com)

Last modified September 2011

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**ABBREVIATIONS**

## Contents of this file

The Hermitian structure of Leech over sqrt(-7) used to construct the extremal 72
dimensional unimodular lattice Gamma commutes with a suitable
structure of Leech over sqrt(5).

Let F be some Gram matrix of the Leech lattice and F5=(5+\sqrt{5})/2 F.
Then the lattice with Gram matrix F5 has minimum norm 8 and hence is an
extremal 5-modular lattice of dimension 24.
A short computer calculation shows that
the Turyn construction applied to F5 yields a 5-modular lattice of
minimum 16. Note that this is the same lattice as Gamma, we just
changed the quadratic form from F^3 to F5^3.
The lattices Gamma with respect to (nF^3+F5^3) hence define
modular lattices of level (n^2+5n+5) and minimum 16+8n (n=0,1,2, ...).
In the limit the density of these lattices tends to the density of
the extremal even unimodular lattice.
A computation with Hilbert modular forms shows that the
Hilbert Theta Series of the lattice Gamma
over Z[(1+sqrt(5))/2] is uniquely determined by the fact that the
unimodular lattice (Gamma,F^3) has minimum 8.
In particular all 75411000 minimal vectors of (Gamma,F5^3) are
also minimal vectors of (Gamma,F^3), so the kissing number of all
lattices (Gamma, nF^3+F5^3) is 75411000 (whereas the one of
(Gamma,F^3) is 6218175600.
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Haftungsausschluss/Disclaimer

Gabriele Nebe