## Computations of cyclotomic lattices.(English)Zbl 0873.11026

The authors study certain modular lattices of level $$l$$ and dimension $$2(p-1), p$$ a prime, using a construction involving the ideal class group of the $$p$$th cyclotomic extension of $$\mathbb Q(\sqrt{-l}).$$ In particular, a family of even, hermitian-unimodular lattices related to the Craig lattices are obtained making use of $$\text{Gal} (\mathbb Q(\sqrt{-l})/\mathbb Q)$$-invariant ideal classes. In the course of their computational investigations, aided by PARI, several new (and some old) examples of extremal lattices are produced in dimensions 12, 20, 24, 32, 36 and 56, and the question of eutaxy is also decided.
Reviewer: K.Roegner (Berlin)

### MSC:

 11E12 Quadratic forms over global rings and fields 11H55 Quadratic forms (reduction theory, extreme forms, etc.) 11R18 Cyclotomic extensions 11-04 Software, source code, etc. for problems pertaining to number theory

PARI/GP
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### References:

 [1] Batut C., User’s Guide to Pari-GP. [2] Bachoc C., Exper. Math. 1 pp 184– (1992) · Zbl 0787.11024 [3] Bayer-Fluckiger E., J. reine angew. Math. 451 pp 51– (1994) [4] Conway J. H., Sphere Packings, Lattices and Groups (1988) · Zbl 0634.52002 [5] DOI: 10.1112/plms/s3-29.4.633 · Zbl 0312.20003 [6] Lang S., Cyclotomic Fields I and II (1990) · Zbl 0704.11038 [7] Ozeki M., Théorie des nombres pp 772– (1989) [8] Plesken W., Memoirs Amer. Math. Soc. 116 pp 1– (1995) [9] Quebbemann H.-G., Math. Nachr. 156 pp 219– (1992) · Zbl 0774.11018 [10] Quebbemann H.-G., J. Number Theory 54 (1995) [11] Scharlau R., ”Classification of integral lattices with large class number” · Zbl 0919.11031
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