Minkowski-Siegel masses of the genera of unimodular lattices

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Numbers

$\varepsilon$

$n$

$m_{\varepsilon}(n)$

Type I (odd)

1/2

Type I (odd)

1/8

Type I (odd)

1/48

Type I (odd)

1/384

Type I (odd)

1/3840

Type I (odd)

1/46080

Type I (odd)

1/645120

Type I (odd)

1/10321920

comment: The only odd class in this dimension is $\mathbb{Z}^8$.

Type I (odd)

17/2786918400

Type I (odd)

10:

1/2229534720

Type I (odd)

11:

31/735746457600

Type I (odd)

12:

31/5885971660800

Type I (odd)

13:

691/765176315904000

Type I (odd)

14:

42151/192824431607808000

Type I (odd)

15:

29713/385648863215616000

Type I (odd)

16:

505121/12340763622899712000

Type I (odd)

17:

642332179/18881368343036559360000

Type I (odd)

18:

692319119/15105094674429247488000

Type I (odd)

19:

8003636403977/77489135679822039613440000

Type I (odd)

20:

248112728523287/619913085438576316907520000

Type I (odd)

21:

593468652605200909/216969579903501710917632000000

Type I (odd)

22:

50904295073459007001/1507367607750643465322496000000

Type I (odd)

23:

1015740532498234470066371/1317439289174062388691861504000000

Type I (odd)

24:

701876707956280018815862361/21079028626784998219069784064000000

Type I (odd)

25:

84715059480304651623612272842147/30465396080006318014267329085440000000

Type I (odd)

26:

14616335635894388876188472684851927/31871491283698917307233513504768000000

Type I (odd)

27:

1894352751772146867430486995462923265007/12429881600642577749821070266859520000000

Type I (odd)

28:

10345060377427694043037889482223023950203227/99439052805140621998568562134876160000000

Type I (odd)

29:

4285009823959590682115628739356169586687220752159/28837325313490780379584883019114086400000000

Type I (odd)

30:

156429914319579070270102710292957201465725850451195039/346047903761889364555018596229369036800000000

Type I (odd)

31:

447543572700878404232772149927275573042725639059585587715489/150184790232659984216878070763546161971200000000

Type I (odd)

32:

416022631331982837344253774635747627157753429574979915187099394241/9611826574890238989880196528866954366156800000000

Type II (even)

comment: The zero-dimensional lattice has trivial isometry group.

Type II (even)

1/696729600

comment: The genus consists of the single isometry class $E_8$.

Type II (even)

16:

691/277667181515243520000

comment: The two classes are $E_8\oplus E_8$ and $D_{16}^{+}$.

Type II (even)

24:

1027637932586061520960267/129477933340026851560636148613120000000

comment: This is the sum of reciprocal full automorphism orders of the 24 Niemeier lattices.

Type II (even)

32:

4890529010450384254108570593011950899382291953107314413193123/121325280941552041649762780685623131486814208000000000

Definition

The mass of the genus of positive definite integral unimodular lattices of dimension $n$ and parity $\varepsilon$ is $m_{\varepsilon}(n)=\sum_{[L]\in\mathcal G_{\varepsilon,n}}1/|O(L)|$. The sum is over isometry classes, and $O(L)$ is the full integral isometry group [3].

Parameters

$\varepsilon$

— parity of the genus

$n$

— dimension ($n\geq1$ for type I; $n\geq0$ and $8\mid n$ for type II)

Formulas

(1)

Let $P_j=\prod_{i=1}^{j}|B_{2i}|$ and $P_0=1$, where $B_r$ are the Bernoulli numbers with $B_1=-1/2$. The signed Euler numbers $E_r$ are defined by $1/\cosh t=\sum_{r\geq0}E_r t^r/r!$, so $E_0=1$, $E_2=-1$, and $E_4=5$.

(2)

For $n=2k+1>1$, $m_{\mathrm I}(n)=\dfrac{2^k+\sigma_n}{2^{2k+1}k!}P_k$, where $\sigma_n=1$ for $n\equiv1,7\pmod8$ and $\sigma_n=-1$ for $n\equiv3,5\pmod8$. Also $m_{\mathrm I}(1)=1/2$ [1] [2].

(3)

For $n=2k\equiv2,6\pmod8$, $m_{\mathrm I}(n)=\dfrac{|E_{k-1}|P_{k-1}}{2^{2k+1}(k-1)!}$. For $n=2k>0$ divisible by $4$, $m_{\mathrm I}(n)=\dfrac{(1-2^{-k})(1+\tau_n2^{1-k})}{2\,k!}|B_k|P_{k-1}$, where $\tau_n=1$ if $n\equiv0\pmod8$ and $\tau_n=-1$ if $n\equiv4\pmod8$ [1].

(4)

For $n=2k>0$ divisible by $8$, $m_{\mathrm{II}}(n)=\dfrac{|B_k|}{2k}\prod_{j=1}^{k-1}\dfrac{|B_{2j}|}{4j}$ [2].

(5)

For $n>0$ divisible by $8$, $2m_{\mathrm I}(n)=(2^{n/2}-1)(2^{n/2-1}+1)m_{\mathrm{II}}(n)$.

(6)

For $1\leq n\leq8$, $m_{\mathrm I}(n)=1/(2^n n!)$, since $\mathbb Z^n$ is the only odd class and its group consists of signed coordinate permutations [1].

Comments

(7)

A Gram matrix $G$ is positive definite, integral and has determinant $1$. The squared length of a vector with coordinates $x$ is $x^{\mathsf T}Gx$. Type II means that every squared length is even; type I means that some squared length is odd. The parity parameter identifies the genus whose mass is taken. Two lattices lie in the same genus if their forms are isometric over $\mathbb R$ and over $\mathbb Z_p$ for every prime $p$. In positive dimension there is one odd unimodular genus in each dimension and one even unimodular genus exactly when $8\mid n$ [2].

(8)

The group is $O(L)=\{U\in\mathrm{GL}_n(\mathbb Z):U^{\mathsf T}GU=G\}$. It includes central inversion and has no determinant restriction. These are the same full groups as in the classical lattice automorphism orders. In dimension zero the single lattice is even and has trivial isometry group, so $m_{\mathrm{II}}(0)=1$.

(9)

The rank-one odd genus contains only $\mathbb Z$, whose isometry group is $\{1,-1\}$. Thus $m_{\mathrm I}(1)=1/2$. Theorem 2 and Table 2 of [1] give $1$ at this exceptional endpoint; the full-group definition gives $1/2$, as also recorded in Table 6 of [2] (Table 3, p. 18, in the arXiv version).

(10)

The even genus in dimension $8$ contains only $E_8$, whose full automorphism order is $696729600$. In dimension $16$ the two classes are $E_8\oplus E_8$ and $D_{16}^{+}$, with group orders $2\cdot696729600^2$ and $2^{15}16!$. Here $D_{16}^{+}=D_{16}\cup((1/2,\ldots,1/2)+D_{16})$ and $D_{16}=\{x\in\mathbb Z^{16}:\sum_i x_i\equiv0\pmod2\}$. The even genus in dimension $24$ consists of the $24$ Niemeier classes, including the Leech lattice, whose full automorphism orders supply its mass by reciprocal summation [1].

Programs

(P1)

Sage

# The general genus-mass routine uses the full isometry group.
from sage.quadratic_forms.genera.genus import Genus, genera

def unimodular_mass(parity, n):
    n = ZZ(n)
    if parity == "I" and n >= 1:
        return Genus(identity_matrix(ZZ, n)).mass()
    if parity == "II" and n >= 0 and n % 8 == 0:
        return QQ(1) if n == 0 else genera((n, 0), 1, even=True)[0].mass()
    raise ValueError("No such unimodular genus")

unimodular_mass("I", 8)   # 1/10321920
unimodular_mass("II", 8)  # 1/696729600

References

[1]

J. H. Conway and N. J. A. Sloane, The unimodular lattices of dimension up to 23 and the Minkowski-Siegel mass constants, European Journal of Combinatorics 3 (1982), 219–231. Theorem 2 and Tables 2–4. (doi)

[2]

O. D. King, A mass formula for unimodular lattices with no roots, Mathematics of Computation 72 (2003), 839–863. Sections 1–2 and Table 6; in the arXiv version, see Table 3, p. 18. (arXiv) (doi)

Links

[3]

Conway and Sloane, Theorem 2 and Tables 2–4 (author copy)

[4]

SageMath reference: genus mass from local genus symbols

Similar tables

Orders of the automorphism groups of the classical lattices — full isometry orders whose reciprocals give the masses of the single-class cubic and E8 genera

Orders of the automorphism groups of the Niemeier lattices — the reciprocals of its 24 orders sum to the even mass in dimension 24

Bernoulli_numbers — exact Bernoulli factors in the mass formulas

Data properties

Entries are of type: rational number

How well the digits are known: exact

Table is complete: no (it holds every odd genus with $1\leq n\leq32$ and every even genus with $0\leq n\leq32$; this is the range of the classical exact mass tables, including the genera of $E_8$, the two even dimension-16 lattices, the Niemeier lattices, and the large dimension-32 masses)

How they were obtained:

The generator evaluates the Bernoulli and Euler product formulas in rational arithmetic. It computes Bernoulli and signed Euler numbers by their defining recurrences, without floating-point division. The dimension-zero even mass and dimension-one odd mass are obtained directly from their trivial and order-two isometry groups, respectively.

Every one of the 37 values agrees exactly with SageMath's independent general genus-mass routine [4], after controls in dimensions 1, 2 and 8. The 36 values outside the rank-one exception agree with the separately transcribed exact fractions in Tables 2 and 4 of [1]. Class-sum checks cover odd dimensions 1 through 11 and even dimensions 0, 8, 16 and 24; the last sum uses all 24 stored Niemeier orders. The odd/even mass relation is checked at every positive dimension divisible by 8 in the table. Doubling the E8 mass and omitting the Leech class are rejected by the controls. These comparisons are repeated on the values read back from the draft. The attached generator and checks reproduce the calculations.