Tag 028S

\(\Lambda_{24}\)

Lambda24·Leech·Leech lattice

Invariants

Rank24
Signature \((n_+, n_-)\)(24, 0)
Determinant\(1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{24}\).

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)
\(e_{1}\)84444442444242224222000-3
\(e_{2}\)44222222222222112112100-1
\(e_{3}\)42422222222221212211100-1
\(e_{4}\)42242222222221122121100-1
\(e_{5}\)42224222222122222222100-1
\(e_{6}\)42222422222122112121000-1
\(e_{7}\)42222242222121212112000-1
\(e_{8}\)222222241112122212222001
\(e_{9}\)42222221422222222222111-1
\(e_{10}\)42222221242222112211010-1
\(e_{11}\)42222221224221212121001-1
\(e_{12}\)222211122224122212222111
\(e_{13}\)42222221222142222111111-1
\(e_{14}\)221122122212242212222211
\(e_{15}\)212121222122224212222121
\(e_{16}\)211221122112222412222111
\(e_{17}\)42222221222121114222111-1
\(e_{18}\)212121122212122224222211
\(e_{19}\)211222122122122222422121
\(e_{20}\)221121222112122222242111
\(e_{21}\)011110021002122212224222
\(e_{22}\)000000001101121112112422
\(e_{23}\)000000001011112111212242
\(e_{24}\)-3-1-1-1-1-1-11-1-1-11-1111-11112224
 
[[8, 4, 4, 4, 4, 4, 4, 2, 4, 4, 4, 2, 4, 2, 2, 2, 4, 2, 2, 2, 0, 0, 0, -3], [4, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 1, 1, 2, 1, 0, 0, -1], [4, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 1, 2, 2, 1, 1, 1, 0, 0, -1], [4, 2, 2, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 2, 2, 1, 2, 1, 1, 0, 0, -1], [4, 2, 2, 2, 4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 0, 0, -1], [4, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 1, 2, 2, 1, 1, 2, 1, 2, 1, 0, 0, 0, -1], [4, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 1, 2, 0, 0, 0, -1], [2, 2, 2, 2, 2, 2, 2, 4, 1, 1, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 0, 0, 1], [4, 2, 2, 2, 2, 2, 2, 1, 4, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 1, -1], [4, 2, 2, 2, 2, 2, 2, 1, 2, 4, 2, 2, 2, 2, 1, 1, 2, 2, 1, 1, 0, 1, 0, -1], [4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 4, 2, 2, 1, 2, 1, 2, 1, 2, 1, 0, 0, 1, -1], [2, 2, 2, 2, 1, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 1], [4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 2, 2, 2, 2, 1, 1, 1, 1, 1, 1, -1], [2, 2, 1, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 4, 2, 2, 1, 2, 2, 2, 2, 2, 1, 1], [2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 4, 2, 1, 2, 2, 2, 2, 1, 2, 1], [2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 1, 2, 2, 2, 2, 4, 1, 2, 2, 2, 2, 1, 1, 1], [4, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 1, 2, 1, 1, 1, 4, 2, 2, 2, 1, 1, 1, -1], [2, 1, 2, 1, 2, 1, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 2, 4, 2, 2, 2, 2, 1, 1], [2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 2, 4, 2, 2, 1, 2, 1], [2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 4, 2, 1, 1, 1], [0, 1, 1, 1, 1, 0, 0, 2, 1, 0, 0, 2, 1, 2, 2, 2, 1, 2, 2, 2, 4, 2, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 4, 2, 2], [0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 2, 4, 2], [-3, -1, -1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 2, 2, 2, 4]]

Integral lattice

Parityeven
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Nikulin invariants \((r, a, \delta)\)\((24, 0, 0)\)
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{12} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 23)\, \zeta^\Sigma(s - 12)}{\zeta^\Sigma(s - 11)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 23)}{\zeta^\Sigma(s - 11)}\)

Definite lattice

Minimum of \(b(x, x)\), \(x \neq 0\)\(4\)
Kissing number196560

Theta series

\[ \sum_{x \in L} q^{b(x, x)} = 1 + 196560q^{4} + O(q^{5}) \]

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)no
Number of roots0
\(R(L) = \mathbb{Z}\Phi(L)\)\(0\)
Rank of \(R(L)\)0
\([L : R(L)]\)infinite: the rank of \(R(L)\) is less than the rank of \(L\)
\(R(L)\) is primitive in \(L\)yes

Notes

The components \(b(e_i, e_j)\) are those of the section GRAM of the entry Leech of the Catalogue of Lattices.

The catalogue entry gives this reference text: SPLAG p. 131 etc.

Morphisms

References

  1. G. Nebe and N. J. A. Sloane, Catalogue of Lattices, entry Leech. https://www.math.rwth-aachen.de/~Gabriele.Nebe/LATTICES/Leech.html