A005134
Number of n-dimensional unimodular lattices (or quadratic forms).
Original entry on oeis.org
1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 5, 8, 9, 13, 16, 28, 40, 68, 117, 297, 665, 2566, 17059, 374062
Offset: 0
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
- N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
- Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
- Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
- Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See page 854.
a(26)-a(28) added from Bill Allombert's and Gaëtan Chenevier's computations by
Robin Visser, Jan 24 2025
A054911
Number of n-dimensional odd unimodular lattices (or quadratic forms).
Original entry on oeis.org
0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, 9, 13, 16, 28, 40, 68, 117, 273, 665, 2566, 17059, 374062
Offset: 0
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
- Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
- Gaëtan Chenevier, Unimodular hunting, Cogent Seminar, Jul 05 2021.
- Gaëtan Chenevier, Unimodular hunting, Modular Forms Workshop, Oberwolfach online, Feb 2021.
- Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
- Steven R. Finch, Minkowski-Siegel mass constants [Broken link]
- Steven R. Finch, Minkowski-Siegel mass constants
- Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863. See Table 3 page 854.
-
function a(n)
if n lt 3 then return Min(1,n); end if;
L := NumberFieldLattice(QNF(), n);
return #GenusRepresentatives(L);
end function; // Robin Visser, Jan 24 2025
A054909
Number of 8n-dimensional even unimodular lattices (or quadratic forms).
Original entry on oeis.org
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
The classical mass formula shows that the next term is at least 8*10^7.
Oliver King and Richard Borcherds (reb(AT)math.berkeley.edu) have recently improved this estimate and have shown that a(4), the number in dimension 32, is at least 10^9 (Jul 22 2000)
A054908
Number of n-dimensional odd unimodular lattices (or quadratic forms) containing no vectors of norm 1.
Original entry on oeis.org
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 4, 3, 12, 12, 28, 49, 156, 368, 1901, 14493, 357003
Offset: 0
- J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 49.
a(26)-a(28) added from Bill Allombert's and Gaëtan Chenevier's computations by
Robin Visser, Jan 24 2025
A380746
Number of n-dimensional indecomposable unimodular lattices (or quadratic forms).
Original entry on oeis.org
1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 2, 1, 4, 3, 11, 12, 27, 48, 176, 367, 1896, 14489, 356988
Offset: 1
For n = 1, the only 1-dimensional indecomposable unimodular lattice is Z, thus a(1) = 1.
For n = 8, the only 8-dimensional indecomposable unimodular lattice is E8, thus a(8) = 1.
For n = 12, the only 12-dimensional indecomposable unimodular lattice is D12+, thus a(12) = 1.
- Fu Zu Zhu, Construction of nondecomposable positive definite unimodular quadratic forms. Sci. Sinica Ser. A, 30 (1987), no. 1, 19-31.
- Fu Zu Zhu, On nondecomposability and indecomposability of quadratic forms, Sci. Sinica Ser. A, 31 (1988), no. 3, 265-273.
- Bill Allombert and Gaëtan Chenevier, Unimodular Hunting II, arXiv:2410.19569 [math.NT], 2024.
- Etsuko Bannai, Positive definite unimodular lattices with trivial automorphism groups, Mem. Amer. Math. Soc., 85 (1990), no. 429, iv+70 pp.
- Gaëtan Chenevier, Unimodular Hunting, arXiv:2410.18788 [math.NT], 2024.
- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Third edition, Springer-Verlag, New York, 1999. lxxiv+703 pp.
- Oliver D. King, A mass formula for unimodular lattices with no roots, Math. Comp., 72 (2003), no. 242, 839-863.
- O. T. O'Meara, The construction of indecomposable positive definite quadratic forms, J. Reine Angew. Math., 276 (1975), 99-123.
- Wilhelm Plesken, Additively indecomposable positive integral quadratic forms, J. Number Theory, 47 (1994), no. 3, 273-283.
Showing 1-5 of 5 results.
Comments