Fields
Each lattice is stored as lattices/<TAG>.md; these fields are its YAML front matter.
A field that exists only under a hypothesis on \(b\) sits under the key named for that hypothesis,
and that key is present exactly when the hypothesis holds.
Every lattice
| Field | Required | Meaning |
|---|---|---|
tag |
yes | Permanent identifier: four characters from 0-9 and A-Z. The file name is the tag. |
name |
yes | Name as plain text, for search. |
latex |
yes | Name as TeX, without math delimiters. |
aliases |
no | Other names, as plain text. |
rank |
yes | Rank of the underlying free module. |
gram_tensor |
yes | Components of the Gram tensor, the symmetric (0,2)-tensor \(b\), in a basis \(e_1, \dots, e_n\): row \(i\) lists \(b(e_i, e_1), \dots, b(e_i, e_n)\). Values are integers or strings p/q. |
signature |
yes | [n_plus, n_minus]: the numbers of \(v\) with \(b(v, v) > 0\) and with \(b(v, v) < 0\) in a \(b\)-orthogonal basis of \(L \otimes \mathbb{Q}\). They do not depend on the basis (Sylvester's law of inertia). |
determinant |
yes | \(\det(b(e_i, e_j))\). It is the same for every basis of \(L\). |
definiteness |
yes | positive_definite or negative_definite when \(b(x, x)\) has one sign on nonzero \(x\); indefinite when it takes both signs; positive_semidefinite when \(b(x, x) \geq 0\) for all \(x\), \(b \neq 0\) and the determinant is zero, and negative_semidefinite with \(\leq\); zero when \(b = 0\). |
families |
no | Named families that contain the lattice. |
related |
no | Related lattices in the catalogue. |
references |
no | Literature for the lattice. |
integral |
no | Invariants of a lattice whose form takes integer values. Required when every \(b(e_i, e_j)\) is an integer. |
definite |
no | Invariants of a definite lattice. They are stated for a positive definite form; on a negative definite lattice they are the invariants of \(-b\). |
indefinite |
no | Invariants of a lattice on which \(b(x, x)\) takes both signs. |
hyperbolic |
no | Invariants of a nondegenerate lattice of rank at least 2 whose signature is \((1, n)\) or \((n, 1)\). |
root_span |
no | \(\mathbb{Z}\Phi(L)\) for a lattice that is not definite. Present only when \(b\) is not definite. |
root_sublattice |
no | The root sublattice \(R(L) = \mathbb{Z}\Phi(L)\) as a sublattice of \(L\). Present when \(L\) is definite or has a root_span block. |
Integral lattice integral
Invariants of a lattice whose form takes integer values. Required when every \(b(e_i, e_j)\) is an integer.
| Field | Required | Meaning |
|---|---|---|
parity |
yes | even when \(b(x, x)\) is even for every \(x\), odd otherwise. |
level |
no | Least positive \(k\) for which \(k b(x,x)\) is even for every \(x\) in the dual lattice. Requires nonzero determinant. |
modular_scale |
no | The \(k\) of an isometry \(L\cong L^*(k)\) recorded by a dual-isometry morphism. |
discriminant_group |
no | Invariant factors \(d_1 \mid d_2 \mid \dots\), each greater than 1, of the cokernel of the correlation \(L \to \operatorname{Hom}(L, \mathbb{Z})\). The empty list is the trivial group. Required when the determinant is not zero. |
discriminant_sequence |
no | For a nondegenerate even lattice, generators of \(O(A_L,q_L)\), the image of the discriminant action in the stated basis, and representatives of the pointed coset set; full generators of \(O(L)\) are named self-isometry morphisms when available. \(MM(L)=O(A_L,q_L)/\operatorname{im}\rho_L\). A multiplication table and generator cosets are present when the image is normal. |
overlattice_count |
no | The number of integral lattices \(M\) with \(L \subseteq M \subseteq L^*\), with \(M = L\) counted: the number of subgroups \(H\) of the discriminant group \(A_L = L^*/L\) with \(b_{A_L}(H, H) = 0\), for the form \(b_{A_L}(x + L, y + L) = b(x, y) + \mathbb{Z}\) with values in \(\mathbb{Q}/\mathbb{Z}\). Subgroups are counted, not their orbits under the isometries of \(L\). An even \(L\) can have odd lattices among the \(M\). Stated when the determinant is not zero and \(A_L\) has at most 100000 subgroups; otherwise absent, and the count is not decided. |
delta |
no | Nikulin's invariant \(\delta\) of an even lattice with \(2 A_L = 0\): 0 when \(b(x, x)\) is an integer for every \(x\) in the dual lattice \(L^*\), and 1 otherwise. With the rank \(r\) and \(A_L \cong (\mathbb{Z}/2)^a\) it gives \((r, a, \delta)\). Required for, and only for, an even lattice whose determinant is not zero and whose invariant factors all equal 2. |
bad_reduction_primes |
no | The primes that divide \(2 \det L\), in increasing order: the primes \(p\) at which \(Q(x) = b(x, x)\) is degenerate modulo \(p\). Outside them and the primes that divide \(n\), the scheme \(Q(x) = n\) over \(\mathbb{Z}\) has good reduction. Required for, and only for, a nonzero determinant. |
quadratic_character |
no | For rank \(2m\): the discriminant \(d\) of the field \(\mathbb{Q}(\sqrt{D})\) with \(D = (-1)^m \det L\), and 1 when \(D\) is a square. For \(p \nmid 2 \det L\) the Kronecker symbol \(\chi_D(p) = (d / p)\) is 1 exactly when \(Q\) modulo \(p\) is a sum of \(m\) hyperbolic planes, and it determines the number of points of \(Q(x) = n\) over \(\mathbb{F}_{p^k}\). Required for, and only for, an even rank and a nonzero determinant. |
genus_symbol |
no | I (odd) or II (even) with the signature, such as II_{1,9}. When the determinant is not 1 or -1, the local symbol at each prime that divides twice the determinant follows in parentheses, in the notation that SageMath prints. Requires a nonzero determinant. |
genus_class_count |
no | The class number of the genus of \(L\): the number of isometry classes of lattices in the genus, \(L\) counted. Computed by latticedb certify with Genus(G).representatives() of SageMath; for an indefinite binary form, the representatives counted up to equivalence. Requires a nonzero determinant; absent when it is not computed. |
spinor_genus_count |
no | The number of spinor genera in the genus of \(L\), a power of 2 (Conway and Sloane, SPLAG, Chapter 15, Section 9.1): the order of the quotient of the spinor operators by the spinor kernel of Theorems 16 and 17 there, enlarged by the spinor operator of one improper isometry, so that a spinor genus is a union of isometry classes. Computed by latticedb certify with Genus(G).spinor_generators(proper=False) of SageMath. Requires a nonzero determinant and rank at least 3. |
spinor_genera |
no | The number of isometry classes in each spinor genus of the genus of \(L\): first in the spinor genus of \(L\), then in the others in decreasing order. The first entry is the class number of the spinor genus of \(L\), and the sum is the class number of the genus. An indefinite \(L\) has one class in each spinor genus (Eichler; SPLAG, Chapter 15, Theorem 14). For a definite \(L\), latticedb certify iterates \(p\)-neighbours from one lattice of each spinor genus at a prime \(p\) whose spinor operator is in the spinor kernel, so that each neighbour stays in its spinor genus (SPLAG, Chapter 15, Theorem 15), and stores the counts only when the masses \(\sum 1/|O(M)|\) of the classes found add up to the mass of the genus. Requires a nonzero determinant and rank at least 3. |
hyperbolic_index |
no | The largest \(n\) with \(L \cong U^n \oplus L'\) for a lattice \(L'\), where \(U\) is the hyperbolic plane. For an integral \(L\) this is the largest \(n\) with an embedding \(U^n \hookrightarrow L\), because a unimodular sublattice \(M\) of \(L\) satisfies \(L = M \oplus M^{\perp}\). Computed by latticedb certify: \(L \cong U^n \oplus L'\) holds exactly when the genus of \(L\) is the sum of the genus of \(U^n\) and a genus of signature \((n_+ - n, n_- - n)\), since a lattice \(U \oplus L'\) of rank at least 3 is alone in its genus (Nikulin 1980, Theorem 1.13.1*). Requires a nonzero determinant; absent when it is not computed. |
primitive_orbits |
no | For a group \(\Gamma \subseteq O(L)\), the series \(F_{L,\Gamma}(z, w)\) whose coefficient \(c_\Gamma(n)\) is the number of \(\Gamma\)-orbits of primitive vectors \(v\) with \(b(v, v) = n\). The keys are O, SO, O+, SO+, Otilde, SOtilde, Otilde+ and SOtilde+: \(S\) is the kernel of the determinant, \(+\) the kernel of the real spinor norm, and \(\widetilde{O}(L)\) the kernel of \(O(L) \to O(A_L)\). Computed by latticedb certify through \(z^4\) and \(w^4\) for a definite lattice, and from \((A_L, q_L)\) for an even lattice of hyperbolic index at least 2 (Gritsenko, Hulek and Sankaran 2009, Proposition 3.3(i); Nikulin 1980, Theorem 1.14.2); stated with a reference otherwise. Requires a nonzero determinant. |
Series of orbits of primitive vectors integral.primitive_orbits.<group>
The series \(F_{L,\Gamma}(z, w) = c_\Gamma(0) + \sum_{n \geq 1} c_\Gamma(n) z^n + \sum_{n \geq 1} c_\Gamma(-n) w^n\) in \(\mathbb{Z}[[z, w]]\). \(c_\Gamma(n)\) is the number of \(\Gamma\)-orbits on the primitive vectors \(v\) with \(b(v, v) = n\). A coefficient that is not known is null.
| Field | Required | Meaning |
|---|---|---|
constant |
no | \(c_\Gamma(0)\), the number of orbits of primitive isotropic vectors. |
z |
no | \(c_\Gamma(1), c_\Gamma(2), \dots\): the coefficients of \(z, z^2, \dots\) |
w |
no | \(c_\Gamma(-1), c_\Gamma(-2), \dots\): the coefficients of \(w, w^2, \dots\) |
reference |
no | The source of the coefficients; absent when latticedb certify computes them. |
Definite lattice definite
Invariants of a definite lattice. They are stated for a positive definite form; on a negative definite lattice they are the invariants of \(-b\).
| Field | Required | Meaning |
|---|---|---|
minimum |
yes | Least value of \(b(x, x)\) over nonzero \(x\). |
kissing_number |
yes | Number of \(x\) with \(b(x, x)\) equal to the minimum; finite because \(b\) is definite. |
automorphism_group_order |
no | Order of \(O(L)\), computed by latticedb certify with qfauto of PARI/GP. |
minimal_vectors |
no | A complete minimal shell in the record basis; its size equals kissing_number. |
perfect |
no | Whether the rank-one tensors \(v v^T\) of minimal vectors span \(\operatorname{Sym}^2(\mathbb Q^n)\). |
regular |
no | For an integral ternary form: every positive integer represented by its genus is represented by this lattice. |
spinor_regular |
no | For an integral ternary form: every positive integer represented by its spinor genus is represented by this lattice. |
theta_series |
no | Entry \(k\) is the number of \(x\) with \(b(x, x) = k\), for \(k = 0, 1, 2, \dots\) Required for, and only for, integer values. The entries run through \(k = \max(\mu, N)\) at least, where \(\mu\) is the minimum and \(N\) is 12, 8, 6 or 4 for rank at most 4, at most 8, at most 12, or greater. |
root_system |
no | ADE type of \(\Phi_{\{2\}}(L) = \{r \in L : b(r, r) = 2\}\), as irreducible components in any order: [E8], [A1, A1], or [] when it is empty. By Witt's theorem (Conway and Sloane, *Sphere Packings, Lattices and Groups*, 3rd edition, Chapter 4, §3), \(\mathbb{Z}\Phi_{\{2\}}(L)\) is the orthogonal sum of the root lattices of the components. Required for, and only for, integer values. |
roots |
yes | The root system \(\Phi(L)\) as its irreducible components, each with a base; [] when \(L\) has no roots. |
Component of the root system of a definite lattice definite.roots[]
An irreducible component of the root system \(\Phi(L)\) of a definite lattice, with a base.
| Field | Required | Meaning |
|---|---|---|
type |
yes | Type of the component: A1, B2, C3, D4, E6, E7, E8, F4, G2, and the higher ranks of A to D. B_n has \(2n\) short roots, and C_n has \(2n\) long roots. |
scale |
yes | The \(k \neq 0\) with \(b(r, r) = 2k\) for each short root \(r\) of the component. The simple roots have \(b(\alpha_i, \alpha_j) = k\,(\alpha_i, \alpha_j)\), where \((\alpha_i, \alpha_j)\) is the entry of the symmetrized Cartan matrix of the type, normalized so that \((r, r) = 2\) for a short root \(r\). \(k\) is negative on a negative definite lattice. |
simple_roots |
yes | Row \(i\) lists the coordinates of the simple root \(\alpha_i\) in the chosen basis \(e_1, \dots, e_n\). The simple roots are numbered as SageMath's CartanMatrix numbers them. The rows of all components are a basis of \(\mathbb{Z}\Phi(L)\): they are the matrix of the embedding into \(L\) of the orthogonal sum of the root lattices of the components. |
Indefinite lattice indefinite
Invariants of a lattice on which \(b(x, x)\) takes both signs.
| Field | Required | Meaning |
|---|---|---|
isotropic |
yes | Whether \(b(x, x) = 0\) for some nonzero \(x\). |
Hyperbolic lattice hyperbolic
Invariants of a nondegenerate lattice of rank at least 2 whose signature is \((1, n)\) or \((n, 1)\).
| Field | Required | Meaning |
|---|---|---|
reflective |
yes | Whether the subgroup generated by reflections has finite index in the isometry group. |
Roots of a lattice that is not definite root_span
\(\mathbb{Z}\Phi(L)\) for a lattice that is not definite. Present only when \(b\) is not definite.
| Field | Required | Meaning |
|---|---|---|
roots |
yes | Roots of \(L\) that generate \(\mathbb{Z}\Phi(L)\): each row lists the coordinates of one root in the chosen basis \(e_1, \dots, e_n\). [] when \(L\) has no roots. When the rows generate \(L\), that proves \(L = \mathbb{Z}\Phi(L)\). When they do not, the notes of the record prove that each root of \(L\) is in the sublattice that the rows generate. |
norms |
yes | Entry \(i\) is \(b(r_i, r_i)\) for the root \(r_i\) in row \(i\) of roots. |
summands |
no | Lattices \(M_1(k_1), M_2(k_2), \dots\), each a record with a scale, with \(\mathbb{Z}\Phi(L) \cong M_1(k_1) \oplus M_2(k_2) \oplus \cdots\), an orthogonal sum. Present exactly when embedding is present. Absent when \(L = \mathbb{Z}\Phi(L)\). |
embedding |
no | Matrix of an isometric embedding \(M_1 \oplus M_2 \oplus \cdots \to L\) with image \(\mathbb{Z}\Phi(L)\): the rows list the coordinates of the images of the chosen basis vectors of \(M_1\), then of \(M_2\), and so on. |
Related lattice related[]
| Field | Required | Meaning |
|---|---|---|
tag |
yes | Tag of another lattice in the catalogue. |
relation |
yes | One sentence that states how that lattice is related to this one. |
Reference references[]
| Field | Required | Meaning |
|---|---|---|
citation |
yes | Bibliographic citation as plain text. |
url |
no | Address of the source, when it has one. |
Morphisms between two lattices morphisms/<S>-<T>.md
The front matter of morphisms/<S>-<T>.md: morphisms from the lattice \(S\) to the lattice \(T\) of the corpus.
| Field | Required | Meaning |
|---|---|---|
source |
yes | Tag of the source \(S\). |
target |
yes | Tag of the target \(T\). |
morphisms |
yes | The morphisms \(S \to T\). |
Morphism morphisms[]
A morphism \(\varphi \colon S(c) \to T\) of lattices. A \(\mathbb{Z}\)-linear map with \(b_T(\varphi x, \varphi y) = c \, b_S(x, y)\) for a nonzero integer \(c\), 1 by default. Its matrix is in the chosen bases of \(S\) and \(T\), which list the orthogonal summands in the order of their names.
| Field | Required | Meaning |
|---|---|---|
name |
yes | Name as plain text; TeX between $ signs is rendered. |
description |
no | One or two sentences on the morphism, as plain text with TeX between $ signs. |
matrix |
yes | Matrix of \(\varphi\), with \(\operatorname{rank} T\) rows and \(\operatorname{rank} S\) columns: column \(j\) lists the coordinates of \(\varphi(e_j)\) in the chosen basis of \(T\), so that \(M^{\top} G_T M = c \, G_S\) for the matrices \(G_S = (b_S(e_i, e_j))\) and \(G_T = (b_T(e_i, e_j))\). A SageMath morphism phi gives phi.matrix().transpose(), because SageMath lists the images in rows. |
scale |
no | The nonzero integer \(c\) with \(b_T(\varphi x, \varphi y) = c \, b_S(x, y)\): the map is a morphism \(S(c) \to T\) from the twist of \(S\) by \(c\). Absent for \(c = 1\). A lattice that a source names as a twist \(M(c)\) of the record \(M\) maps to \(T\) with this scale. |
row_subdivisions |
no | Lines between the rows, as M.subdivisions() of SageMath states them: a line \(k\) lies between rows \(k\) and \(k + 1\). The basis vectors of \(T\) between two consecutive lines span an orthogonal summand of \(T\). |
column_subdivisions |
no | Lines between the columns: a line \(k\) lies between columns \(k\) and \(k + 1\). The basis vectors of \(S\) between two consecutive lines span an orthogonal summand of \(S\). |
Properties
| Property | Meaning |
|---|---|
| integral | \(L\) is integral. |
| not integral | \(L\) is not integral. |
| even | \(L\) is integral and even. |
| odd | \(L\) is integral and odd. |
| unimodular | \(L\) is unimodular. |
| p-elementary | \(A_L \cong (\mathbb{Z}/p)^a\) for a prime \(p\) and \(a \geq 1\); the label states the prime, as in 2-elementary. |
| root lattice | \(L = \mathbb{Z}\Phi(L)\). |
| not a root lattice | \(L \neq \mathbb{Z}\Phi(L)\). |
| degenerate | \(L\) is degenerate. |
| hyperbolic | \(L\) is hyperbolic. |
| isotropic | \(L\) is indefinite and isotropic. |
| anisotropic | \(L\) is indefinite and not isotropic. |
| reflective | \(L\) is reflective. |
| not reflective | \(L\) is hyperbolic and not reflective. |
Families
| Family | Lattices | Meaning |
|---|---|---|
diagonal |
76 | The basis is orthogonal: \(b(e_i, e_j) = 0\) for \(i \neq j\). For example \(\langle a \rangle\) and \(\mathrm{I}_{p,q}\). |
irreducible-root-lattice |
39 | An irreducible root lattice \(A_n\), \(D_n\), \(E_6\), \(E_7\) or \(E_8\), in the basis of its simple roots. |
root-lattice-sum |
24 | An orthogonal sum of two or more root lattices \(A_n\), \(D_n\), \(E_6\), \(E_7\), \(E_8\), in the union of their bases of simple roots. |
dual-root-lattice |
39 | The dual lattice \(R^*\) of a root lattice \(R\), in the basis of fundamental weights, dual to the basis of simple roots of \(R\). |
affine-root-lattice |
7 | The lattice of an affine root system \(\widetilde{A}_n\), \(\widetilde{D}_n\), \(\widetilde{E}_6\), \(\widetilde{E}_7\) or \(\widetilde{E}_8\) in the basis of its simple roots: positive semidefinite, with a radical of rank 1. |
even-unimodular |
14 | Even and unimodular: an orthogonal sum \(U^{s} \oplus E_8(-1)^{t}\) with \(s \geq 1\), of signature \((s, s + 8t)\). |
r-plus-a-22 |
12 | An even hyperbolic 2-elementary lattice with Nikulin's invariants \((r, a, \delta)\) and \(r + a = 22\). Nikulin classifies these lattices (J. Soviet Math. 22 (1983), DOI 10.1007/BF01094757, Table 1); Theorem 1.1 of Alexeev, arXiv:2209.09110, decides which of them are reflective. |
laminated |
13 | A laminated lattice \(\Lambda_n\) of Conway and Sloane, in the basis of its entry of the Catalogue of Lattices. |
nebe-sloane-catalogue |
44 | An entry of the Catalogue of Lattices of G. Nebe and N. J. A. Sloane, with the Gram tensor as the entry states it. |
nikulin-two-elementary |
75 | An even hyperbolic lattice \(L\) with \(2 A_L = 0\) of signature \((1, r - 1)\), named by Nikulin's invariants \((r, a, \delta)\): one of the 75 invariant lattices of the nonsymplectic involutions of K3 surfaces in Table 1 of Nikulin, J. Soviet Math. 22 (1983), DOI 10.1007/BF01094757. The 13 with \(a = r\) are the twists \(M(2)\) of unimodular lattices \(M\), and they are the only twists that the corpus records. |
beauville-bogomolov |
10 | \(H^2(Y, \mathbb{Z})\) with the Beauville-Bogomolov form, for \(Y\) an irreducible holomorphic symplectic variety of one of the known deformation types \(S^{[n]}\), \(K_n(T)\), OG6, OG10, as the table of the introduction of Rapagnetta, arXiv:math/0606409, states it. |
polarized-k3 |
12 | The lattice \(L_{2d} \cong U^2 \oplus E_8(-1)^2 \oplus \langle -2d \rangle\), the orthogonal complement in \(\mathrm{II}_{3,19}\) of a primitive vector \(h\) with \(b(h, h) = 2d\); the period domain of K3 surfaces with a polarization of degree \(2d\) is defined from it. |
niemeier |
24 | An even unimodular positive definite lattice of rank 24: one of the 24 lattices of Niemeier's classification, named by its root system; the Leech lattice is the one with no roots. |
k3-symplectic-invariant |
43 | A lattice \(M\) such that \(M(k)\), for a nonzero integer \(k\), is the invariant lattice \(\Lambda^G\) of a finite group \(G\) of symplectic isometries of the K3 lattice \(\Lambda = \mathrm{II}_{3,19}\): a row of Table 10.3 of K. Hashimoto, arXiv:1012.2682. |
k3-symplectic-coinvariant |
40 | A lattice \(M\) such that \(M(k)\), for a positive integer \(k\), is \(\Lambda_G(-1)\), for the coinvariant lattice \(\Lambda_G = (\Lambda^G)^\perp\), negative definite, of a finite group \(G\) of symplectic isometries of the K3 lattice \(\Lambda = \mathrm{II}_{3,19}\): a row of Table 10.2 of K. Hashimoto, arXiv:1012.2682. |