Tag 027M

\(\mathrm{I}_{1,25}\)

I_{1,25}

Invariants

Rank26
Signature \((n_+, n_-)\)(1, 25)
Determinant\(-1\)

Gram tensor

The components \(b(e_i, e_j)\) of the Gram tensor \(b\) in the chosen basis \(e_{1}, \dots, e_{26}\).

In this basis, \(L = {}\)\(\mathbb{Z}\{e_{1}\}\) \(\oplus\) \(\mathbb{Z}\{e_{2}\}\) \(\oplus\) \(\mathbb{Z}\{e_{3}\}\) \(\oplus\) \(\mathbb{Z}\{e_{4}\}\) \(\oplus\) \(\mathbb{Z}\{e_{5}\}\) \(\oplus\) \(\mathbb{Z}\{e_{6}\}\) \(\oplus\) \(\mathbb{Z}\{e_{7}\}\) \(\oplus\) \(\mathbb{Z}\{e_{8}\}\) \(\oplus\) \(\mathbb{Z}\{e_{9}\}\) \(\oplus\) \(\mathbb{Z}\{e_{10}\}\) \(\oplus\) \(\mathbb{Z}\{e_{11}\}\) \(\oplus\) \(\mathbb{Z}\{e_{12}\}\) \(\oplus\) \(\mathbb{Z}\{e_{13}\}\) \(\oplus\) \(\mathbb{Z}\{e_{14}\}\) \(\oplus\) \(\mathbb{Z}\{e_{15}\}\) \(\oplus\) \(\mathbb{Z}\{e_{16}\}\) \(\oplus\) \(\mathbb{Z}\{e_{17}\}\) \(\oplus\) \(\mathbb{Z}\{e_{18}\}\) \(\oplus\) \(\mathbb{Z}\{e_{19}\}\) \(\oplus\) \(\mathbb{Z}\{e_{20}\}\) \(\oplus\) \(\mathbb{Z}\{e_{21}\}\) \(\oplus\) \(\mathbb{Z}\{e_{22}\}\) \(\oplus\) \(\mathbb{Z}\{e_{23}\}\) \(\oplus\) \(\mathbb{Z}\{e_{24}\}\) \(\oplus\) \(\mathbb{Z}\{e_{25}\}\) \(\oplus\) \(\mathbb{Z}\{e_{26}\}\), an orthogonal sum.

\(e_{1}\)\(e_{2}\)\(e_{3}\)\(e_{4}\)\(e_{5}\)\(e_{6}\)\(e_{7}\)\(e_{8}\)\(e_{9}\)\(e_{10}\)\(e_{11}\)\(e_{12}\)\(e_{13}\)\(e_{14}\)\(e_{15}\)\(e_{16}\)\(e_{17}\)\(e_{18}\)\(e_{19}\)\(e_{20}\)\(e_{21}\)\(e_{22}\)\(e_{23}\)\(e_{24}\)\(e_{25}\)\(e_{26}\)
\(e_{1}\)10000000000000000000000000
\(e_{2}\)0-1000000000000000000000000
\(e_{3}\)00-100000000000000000000000
\(e_{4}\)000-10000000000000000000000
\(e_{5}\)0000-1000000000000000000000
\(e_{6}\)00000-100000000000000000000
\(e_{7}\)000000-10000000000000000000
\(e_{8}\)0000000-1000000000000000000
\(e_{9}\)00000000-100000000000000000
\(e_{10}\)000000000-10000000000000000
\(e_{11}\)0000000000-1000000000000000
\(e_{12}\)00000000000-100000000000000
\(e_{13}\)000000000000-10000000000000
\(e_{14}\)0000000000000-1000000000000
\(e_{15}\)00000000000000-100000000000
\(e_{16}\)000000000000000-10000000000
\(e_{17}\)0000000000000000-1000000000
\(e_{18}\)00000000000000000-100000000
\(e_{19}\)000000000000000000-10000000
\(e_{20}\)0000000000000000000-1000000
\(e_{21}\)00000000000000000000-100000
\(e_{22}\)000000000000000000000-10000
\(e_{23}\)0000000000000000000000-1000
\(e_{24}\)00000000000000000000000-100
\(e_{25}\)000000000000000000000000-10
\(e_{26}\)0000000000000000000000000-1
 
[[1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1]]

Integral lattice

Parityodd
Discriminant group\(0\)
Unimodularyes
Integral overlattices \(L \subseteq M \subseteq L^*\)1
Primes of bad reduction \(\Sigma_L = \{p \mid 2 \det L\}\)\(\{2\}\)
Character of the discriminant \(\chi_D\), \(D = (-1)^{13} \det L\)trivial
Zeta function of the cone \(X_0 : Q(x) = 0\), \(\Sigma = \Sigma_L\)\(\zeta^\Sigma(X_0, s) = \frac{\zeta^\Sigma(s - 25)\, \zeta^\Sigma(s - 13)}{\zeta^\Sigma(s - 12)}\)
Zeta function of \(X_n : Q(x) = n\), \(n \neq 0\), \(\Sigma = \Sigma_L \cup \{p \mid n\}\)\(\zeta^\Sigma(X_n, s) = \frac{\zeta^\Sigma(s - 25)}{\zeta^\Sigma(s - 12)}\)

Indefinite lattice

Isotropicyes

Roots

Root lattice: \(L = \mathbb{Z}\Phi(L)\)yes
\(L = \mathbb{Z}\Phi_S(L)\) for\(S = \{-1, 1\}\)
\(R(L) = \mathbb{Z}\Phi(L)\)\(L\)
Rank of \(R(L)\)26
\([L : R(L)]\)1
\(R(L)\) is primitive in \(L\)yes
ProofEach \(r_i\) in roots that generate \(\mathbb{Z}\Phi(L)\) is a root, and the \(r_i\) generate \(L\).

Roots that generate \(\mathbb{Z}\Phi(L)\)

The \(r_i\) are a basis of \(L\).

RootCoordinates\(b(r, r)\)
\(r_{1}\)\(e_{1}\)\(1\)
\(r_{2}\)\(e_{2}\)\(-1\)
\(r_{3}\)\(e_{3}\)\(-1\)
\(r_{4}\)\(e_{4}\)\(-1\)
\(r_{5}\)\(e_{5}\)\(-1\)
\(r_{6}\)\(e_{6}\)\(-1\)
\(r_{7}\)\(e_{7}\)\(-1\)
\(r_{8}\)\(e_{8}\)\(-1\)
\(r_{9}\)\(e_{9}\)\(-1\)
\(r_{10}\)\(e_{10}\)\(-1\)
\(r_{11}\)\(e_{11}\)\(-1\)
\(r_{12}\)\(e_{12}\)\(-1\)
\(r_{13}\)\(e_{13}\)\(-1\)
\(r_{14}\)\(e_{14}\)\(-1\)
\(r_{15}\)\(e_{15}\)\(-1\)
\(r_{16}\)\(e_{16}\)\(-1\)
\(r_{17}\)\(e_{17}\)\(-1\)
\(r_{18}\)\(e_{18}\)\(-1\)
\(r_{19}\)\(e_{19}\)\(-1\)
\(r_{20}\)\(e_{20}\)\(-1\)
\(r_{21}\)\(e_{21}\)\(-1\)
\(r_{22}\)\(e_{22}\)\(-1\)
\(r_{23}\)\(e_{23}\)\(-1\)
\(r_{24}\)\(e_{24}\)\(-1\)
\(r_{25}\)\(e_{25}\)\(-1\)
\(r_{26}\)\(e_{26}\)\(-1\)

Morphisms