Constructions of Dense Lattices over Number Fields

ABSTRACT

In this work, we present constructions of algebraic lattices in Euclidean space with optimal center density in dimensions 2;3;4;5;6;8 and 12, which are rotated versions of the lattices Λ_n , for $n=2,3,4,5,6,8$ and K ₁₂. These algebraic lattices are constructed through canonical homomorphism via ℤ-modules of the ring of algebraic integers of a number field.

Keywords:
algebric lattices; number fields; sphere packings

RESUMO

Neste trabalho, apresentamos construçõoes de reticulados algébricos no espaço euclidiano com densidade central ótima nas dimensões 2, 3, 4, 5, 6, 8 e 12, que são versões rotacionadas dos reticulados Λ_n , para $n=2,3,4,5,6,8$ e K ₁₂, onde esses reticulados algébricos são construídos através do homomorfismo canônico via ℤ-módulos do anel de inteiros algébricos de um corpo de números.

Palavras-chave:
reticulados algébricos; corpos de números; empacotamento esférico

1 INTRODUCTION

Algebraic number theory has recently raised a great interest for its new role in algebraic lattice theory and in code design for many different coding applications. Algebraic lattices have been useful in information theory and the question of finding algebraic lattices over number fields maximum center density. The problem of finding algebraic lattices with maximal minimum product distance has been studied in last years and this has motived special attention of many researchs in considering ideals of certain rings. The search for dense algebraic lattices in general dimensions has been encouraged in the last decades because they can be applied to Information Theory ^{1) - (4}.

The classical sphere packing problem consists to find out how densely a large number of identical spheres can be packed together in the Euclidean space. The packing density, ∆(Λ), of a lattice Λ is the proportion of the space ℝⁿ covered by the non-overlapping spheres of maximum radius centered at the points of Λ. The densest possible lattice packings have only be determined in dimensions 1 to 8 and 24. It is also known that these densest lattice packings are unique (up to equivalences) ⁵.

This paper is organized as follows. In Section 2, notions and results from algebraic number theory that are used in the work are reviewed. In Section 3, rotated lattices are constructed from number fields in dimensions 2,3,4,5,6,8 and 12, which are rotated versions of the lattices Ln, for $n=2,3,4,5,6,8$ and K ₁₂.

2 BACKGROUND OF NUMBER FIELDS

Let 𝕂 be a number field, i.e., 𝕂 is a finite extension of ℚ. By Primitive Element Theorem, there is an element $\theta \in 𝕂$ such that $\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)=\left\{\sum _{i=0}^{\in n-1}{a}_i\mathrm{\mathbb{Q}}\right\}$, where θ is a root of a polynomial $p\left(x\right)\in \mathrm{\mathbb{Q}}\left[x\right]$ of minimal degree n. A cyclotomic field is a number field such that $\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)$, where θ is a primitive n-th root of unity. If ${\theta }_1=\theta ,{\theta }_2,...,{\theta }_n$ are the n distinct roots of p(x), then threre are exactly n distinct ℚ-embeddings ${\sigma }_i:\mathbb{K}\to \mathrm{ℂ}$ such that ${\sigma }_i\left(\theta \right)={\theta }_i$, for all $i=1,2,\dots ,n$. Furhtermore, there are r ₁ real embeddings ${\sigma }_1,\dots ,{\sigma }_{r_1}$ and 2r ₂ complex embeddings ${\sigma }_{r_1+1},\overline{{\sigma }_{r_1+1}},\cdots ,{\sigma }_{r_1+r_2},\overline{{\sigma }_{r_1+r_2}}$. If ℜ(x) and ℑ(x) denote, respectively, the real part and the imaginary part of x, the canonical embedding $\sigma :\mathbb{K}\to {\mathrm{ℝ}}^n$, with $x\in \mathbb{K}$, is defined by

$\sigma \left(x\right)=\left({\sigma }_1\left(x\right),\cdots ,{\sigma }_{r_1}\left(x\right),\mathfrak{R}\left({\sigma }_{r_1+1}\left(x\right)\right),\cdots ,\mathfrak{J}\left({\sigma }_{r_1+r_2}\left(x\right)\right)\right).$

The set $\mathcal{O}\mathbb{K}=\{\alpha :f(\alpha )=0$ for some monic polynomial $f\left(x\right)\in \mathrm{\mathbb{Z}}\left[x\right]\}$ is a ring called ring of algebraic integers of 𝕂. The ring 𝒪 _𝕂 has a basis $\left\{{\alpha }_1,{\alpha }_2,\dots ,{\alpha }_n\right\}$ over ℤ. In other words, every element $\alpha \in {\mathcal{O}}_{\mathbb{K}}$ is uniquely written as $\alpha =\sum _{i=1}^n{a}_i{\alpha }_i$, where ${\alpha }_i\in \mathrm{\mathbb{Z}}$ for all $i=1,2,\dots ,n$, and every nonzero fractional ideal ℳ of 𝒪 _𝕂 is a free ℤ-module of rank n⁷.

If $\alpha \in \mathbb{K}$, the value

$T{r}_{\mathbb{K}}\left(\alpha \right)=\sum _{i=1}^n{\sigma }_i\left(\alpha \right)$

is called trace of α in 𝕂. If $\left\{{\alpha }_1,{\alpha }_2,\cdots ,{\alpha }_n\right\}$ is an integral basis of 𝕂, the discriminant of 𝕂 is defined as $D_{\mathbb{K}}=det{\left[{\sigma }_j\left({\alpha }_i\right)\right]}^2$ and it is an invariant over change of basis ⁶.

3 CONSTRUCTIONS OF DENSE ALGEBRAIC LATTICES

A lattice ∧ is a discrete additive subgroup of ℝⁿ , that is, $\left\{0\right\}\ne \wedge \subseteq {\mathrm{ℝ}}^n$ is a lattice iff there are linearly independent vectors $\left\{v_1,v_2,...,v_k\right\}$, with $k\le n$, in ℝⁿ such that

$\Lambda =\left\{\sum _{i=1}^k{a}_i{v}_i:a_i\in \mathrm{\mathbb{Z}},\text{for all}i=1,2,\cdots ,k\right\}.$

The set $\left\{v_1,v_2,...,v_k\right\}$ is called a basis for ∧, the matrix M whose rows are these vectors is called a generator matrix for ∧ and the matrix $G=M{M}^t$ is called Gram matrix.

If M is a ℤ-submodule in 𝕂 of rank n, the set $\Lambda =\sigma \left(\mathcal{M}\right)$ is a lattice in ℝⁿ called an algebraic lattice. The center density of Λ is given by

$\delta \left(\Lambda \right)=\frac{t^{n/2}}{2^n\left[{\mathcal{O}}_{\mathbb{K}}:\mathcal{M}\right]\sqrt{\left|D_{\mathbb{K}}\right|}},$

where $t=min\left\{T{r}_{\mathbb{K}}\right(\alpha \overline{\alpha }):\alpha \in ℳ,\alpha \ne 0\}$ and $[{𝒪}_{\mathbb{K}}:ℳ]$ denotes the index of the submodule M.

Example 3.1.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left({\zeta }_3\right)$, where ζ₃ is the primitive 3-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=2,\{1,{\zeta }_3\}$is a basis of 𝕂 and$D_{\mathbb{K}}=-3$. If ℳ is a submodule of 𝒪 _𝕂 given by

$\mathcal{M}=\left\{a_0+a_1{\zeta }_3:a_0,a_1\in \mathrm{\mathbb{Z}}\right\},$

then $[{𝒪}_{\mathbb{K}}:ℳ]=1$ and

$T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=2\left(a_0^2-a_0{a}_1+a_1^2\right),$

where $\alpha \in ℳ$ . Since $t=min\left\{T{r}_{\mathbb{K}}\right(aā):\alpha \in ℳ;\alpha \ne 0\}=2$ with $a_0=1$ and $a_1=0$ , it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{2}/2\right)}^2}{\sqrt{3}}=\frac{1}{2\sqrt{3}},$

i.e., the center density of σ(ℳ) is the same of the lattice ∧₄. Similarly, if$\mathbb{K}=\mathrm{\mathbb{Q}}\left(\sqrt{3}\right)$, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=2,\{1,\sqrt{3}\}$is a basis of 𝕂 and$D_{\mathbb{K}}=12$. If ℳ is a submodule of 𝒪 _𝕂 given by$ℳ=\left\{a_0+a_1\sqrt{3}:a_0-a_1\equiv 0(mod2)\text{and}{a}_0,a_1\in \mathrm{\mathbb{Z}}\right\}$, then$[{𝒪}_{\mathbb{K}}:ℳ]=2$and$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=8a_0^2+24a_0{a}_1+24a_1^2$, where$\alpha \in ℳ$. Since$t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=8$with$a_0=1$and$a_1=0$, it follows that$\delta \left(ℳ\right)=\frac{{\left(\sqrt{8}/2\right)}^2}{2^3\sqrt{3}}=\frac{1}{4\sqrt{3}}$.

Example 3.2.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)$, where$\theta ={\zeta }_9+{\zeta }_9^{-1}$and ζ₉is the primitive 9-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=3,\{1,\theta ,{\theta }^2\}$is an integral basis of 𝕂 and$D_{\mathbb{K}}=3^4$. If ℳ is a submodule of 𝒪 _𝕂 given by

$ℳ=\left\{a_0+a_1\theta +a_2{\theta }^2:a_0\equiv 0(mod2)\text{and}{a}_0+2a_1+a_2\equiv 0(mod3),\text{where}{a}_0,a_1,a_2\in \mathrm{\mathbb{Z}}\right\},$

then $[{𝒪}_{\mathbb{K}}:ℳ]=6$ and

$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=18(a_0^2+a_0{a}_1+5a_0{a}_2+a_1^2+5a_1{a}_2+9a_2^2),$

where $\alpha \in ℳ$ . Since $t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=18$ with $a_0=1$ and $a_1=a_2=0$ , it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{18}/2\right)}^3}{54}=\frac{1}{4\sqrt{2}},$

i.e., the center density of σ(ℳ) is the same of the lattice Λ₃ . Similarly, if$ℳ=\left\{a_0+a_1\theta +a_2{\theta }^2:a_0\equiv 0(mod2)\text{and}{a}_0+2a_1+a_2\equiv 0(mod3),\text{where}{a}_0,a_1,a_2\in \mathrm{\mathbb{Z}}\right\}$, then$[{𝒪}_{\mathbb{K}}:ℳ]=6$and$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=18(3a_0^2+3a_0{a}_1+10a_0{a}_2+a_1^2+5a_1{a}_2+9a_2^2)$, where$\alpha \in ℳ$. Since$t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=18$with$a_0=a_2=0$and$a_1=1$, it follows that$\delta \left(ℳ\right)=\frac{{\left(\sqrt{18}/2\right)}^3}{2·3·3^2}\frac{1}{4\sqrt{2}}$. Similarly, if$ℳ=\left\{a_0+a_1\theta +a_2{\theta }^2:a_0\equiv 0(mod2)\text{and}{a}_0+2a_1+a_2\equiv 0(mod3),\text{where}{a}_0,a_1,a_2\in \mathrm{\mathbb{Z}}\right\}$, then$[{𝒪}_{\mathbb{K}}:ℳ]=6$and$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=18(3a_0^2+3a_0{a}_1+10a_0{a}_2+a_1^2+5a_1{a}_2+9a_2^2)$, where$\alpha \in ℳ$. Since$t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=18$with$a_0=a_2=0$and$a_1=1$, it follows that$\delta \left(ℳ\right)=\frac{{\left(\sqrt{18}/2\right)}^3}{2·3·3^2}=\frac{1}{4\sqrt{2}}$. Finally, if$\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)$, where θ is a root of$p\left(x\right)=x^3-3x+1$, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=3,\{1,\theta ,{\theta }^2\}$is a basis of 𝕂 and$D_{\mathbb{K}}=3^4$. If ℳ is a submodule of 𝒪 _𝕂 given by$ℳ=\left\{a_0+a_1\theta +a_2{\theta }^2:a_2\equiv 0(mod2)\text{and}{a}_0-a_1+a_2\equiv 0(mod3),\text{with}{a}_0,a_1,a_2\in \mathrm{\mathbb{Z}}\right\}$, then$[{𝒪}_{\mathbb{K}}:ℳ]=6$and$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=18(a_0^2+5a_0{a}_1+3a_0{a}_2+9a_1^2+10a_1{a}_2+3a_2^2)$, where$\alpha \in ℳ$. Since$t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=18$with$a_0=1$and$a_1=a_2=0$, it follows that$\delta \left(ℳ\right)=\frac{{\left(\sqrt{98}/2\right)}^3}{2·7^3}=\frac{1}{4\sqrt{2}}$.

Example 3.3.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left({\zeta }_8\right)$, where ζ₈is the primitive 8-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=4,\{1,{\zeta }_8,{\zeta }_8^2,{\zeta }_8^3\}$is an integral basis of 𝕂 and$D_{\mathbb{K}}=2^8$. If ℳ is a submodule of 𝒪 _𝕂 given by

$\mathcal{M}=\left\{a_0+a_1{\zeta }_8+a_2{\zeta }_8^2+{a_3}_8^3:a_0+a_1+a_2+a_3\equiv 0\left(mod2\right),\text{where}{a}_0,a_1,a_2,a_3\in \mathrm{\mathbb{Z}}\right\},$

then $[{\mathcal{O}}_{\mathbb{K}}:\mathcal{M}]=2$ and

$T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=8\left(2a_0^2-2a_0{a}_3+a_1^2-a_1{a}_2+a{2}^2-2a_2{a}_3+2a_3^2\right),$

where$\alpha \in \mathcal{M}$. Since$t=min\left\{T{r}_{\mathbb{K}}\left({\alpha }^2\right):\alpha \in \mathcal{M},\alpha \ne 0\right\}=8$with$a_1=1$and$a_0=a_2=a_3=0$, it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{8}/2\right)}^4}{32}=\frac{1}{8},$

i.e., the center density of σ(ℳ) is the same of the lattice Λ₄. Similarly, if$\mathcal{M}=\left\{a_0+a_1{\zeta }_8+a_2{\zeta }_8^2+a_3{\zeta }_8^3:a_2+a_3\equiv 0\left(mod2\right),\text{where}{a}_0,a_1,a_2,a_3\in \mathrm{\mathbb{Z}}\right\}$, then$\left[{\mathcal{O}}_{\mathbb{K}}:\mathcal{M}\right]=2$and$T{r}_{\mathbb{K}}\left({\alpha }^2\right)=8\left(a_0^2+a_1^2+a_2^2+2a_3^2+a_0{a}_2+2a_0{a}_3+a_1{a}_2+2a_2{a}_3\right)$, where$\alpha \in \mathcal{M}$. Since$t=min\left\{T{r}_{\mathbb{K}}\left({\alpha }^2\right):\alpha \in \mathcal{M},\alpha \ne 0\right\}=8$with$a_0=1$and$a_1=a_2=a_3=0$, it follows that$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{8}/2\right)}^4}{32}=\frac{1}{8}$. Similarly, if$\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)$, where θ is a root of$p\left(x\right)=x^4+3x^2+1$, then$\left[\mathbb{K}:\mathrm{\mathbb{Q}}\right]=4$, where$\left\{1,\theta ,{\theta }^2,{\theta }^3\right\}$is an integral basis of 𝕂, $D_{\mathbb{K}}=2^4{\dot{5}}^2$. If ℳ is a submodule of 𝒪 _𝕂 given by$\mathcal{M}=\left\{a_0+a_1\theta +a_2{\theta }^2+a_3{\theta }^3:a_0-2a_1+2a_2-a_3\equiv 0\left(mod5\right),\text{where}{a}_0,a_1,a_2,a_3\in \mathrm{\mathbb{Z}}\right\}$, then$\left[{\mathcal{O}}_{\mathbb{K}}:\mathcal{M}\right]=10$, and if$\alpha \in \mathcal{M}$, then$T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=40a_0^2-40a_0{a}_1+132a_0{a}_2+360a_0{a}_3+20a_1^2-28a_1{a}_2-140a_1{a}_3+158a_2^2+720a_2{a}_3+900a_3^2$. Since$t=min\left\{T{r}_{\mathbb{K}}\left({\alpha }^2\right):\alpha \in \mathcal{M},\alpha \ne 0\right\}=20$with$a_0=a_2=a_3=0$and$a_1=1$, if follows that$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{20}/2\right)}^4}{2^3·5^2}=\frac{1}{8}$.

Example 3.4.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left(\theta \right)$, where$\theta ={\zeta }_{44}^{10}-{\zeta }_{44}^{12}$and ζ₄₄is the primitive 44-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=5,\{1,\theta ,{\theta }^2,{\theta }^3,{\theta }^4\}$is an integral basis of 𝕂 and the discriminant of 𝕂 is 11⁴ . Let ℳ be a submodule of 𝒪 _𝕂 given by

$$\begin{gathered} ℳ=\{a_0+a_1\theta +a_2{\theta }^2+a_3{\theta }^3+a_4{\theta }^4:a_0\equiv 0(mod11),5a_2+a_3\equiv 0(mod11) \\ \text{and}{a}_0+15a_1+11a_2+a_4\equiv 0(mod22),\text{where}{a}_0,a_1,a_2,a_3,a_4\in \mathrm{\mathbb{Z}}\}.. \end{gathered}$$

In this case, ℳ is a submodule of 𝒪 _𝕂 of index $2·{11}^3$ and the trace form of $\alpha \in ℳ$ is given by

$$\begin{gathered} T{r}_{\mathbb{K}/\mathrm{\mathbb{Q}}}\left({\alpha }^2\right)=37752a_0^2+43802a_0{a}_1+79134a_0{a}_2+16456a_0{a}_3+136488a_0{a}_4 \\ +12826a_1^2+46706a_1{a}_2+10406a_1{a}_3+79860a_1{a}_4+44286a_2^2 \\ +26136a_2{a}_3+144716a_2{a}_4+9438a_3^2+30976a_3{a}_4+124388a_4^2. \end{gathered}$$

Thus,$t=min\left\{T{r}_{\mathbb{K}}\right({\alpha }^2):\alpha \in ℳ,\alpha \ne 0\}=242$with$a_0=a_2=a_3=0,a_1=-3$and$a_4=1$. As the volume of the lattice σ (mathcalM) is$\sqrt{\left|D_{\mathbb{K}}\right|}\left[\mathcal{M}:{\mathcal{O}}_{\mathbb{K}}\right]=2·{11}^5$, it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{242}/2\right)}^5}{2·{11}^5}=\frac{1}{8\sqrt{2}},$

i.e., the center density of σ(ℳ) is the same of the lattice Λ₇ .

Example 3.5.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left({\zeta }_9\right)$, where ζ₉is the primitive 9-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=6,\{1,{\zeta }_9,{\zeta }_9^2,{\zeta }_9^3,{\zeta }_9^4,{\zeta }_9^5\}$is an integral basis of 𝕂 and$D_{\mathbb{K}}=-3^9$. If ℳ is a submodule of 𝒪 _𝕂 given by

$$\begin{gathered} \mathcal{M}=\{a_0+a_1{\zeta }_9+a_2{\zeta }_9^2+a_3{\zeta }_9^3+a_4{\zeta }_9^4+a_5{\zeta }_9^5:a_1-a_2+a_4-a_5\equiv 0(mod3), \\ \text{where}{a}_0,a_1,...,a_5\in \mathrm{\mathbb{Z}}\},, \end{gathered}$$

then $[{𝒪}_{\mathbb{K}}:ℳ]=9$ and

$$\begin{gathered} T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=18(a_0^2+a_0{a}_1+a_0{a}_2+a_0{a}_3+2a_0{a}_4+2a_0{a}_5+a_1^2+a_1{a}_3+3a_1{a}_4 \\ +a_2^2+a_2{a}_3+3a_2{a}_5+a_3^2+2a_3{a}_4+2a_3{a}_5+3a_4^2+3a_5^2, \end{gathered}$$

where $\alpha \in ℳ$ . Since $t=min\left\{T{r}_{\mathbb{K}}\right(\alpha \overline{\alpha }):\alpha \in ℳ,\alpha \ne 0\}=18$ with $a_0=1$ and $a_1=a_2=a_3=a_4=a_5=0$ , it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{18}/2\right)}^6}{3^6\sqrt{3}}=\frac{1}{8\sqrt{3}},$

i.e., the center density of σ(ℳ) is the same of the lattice Λ₆ .

Example 3.6.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left({\zeta }_{20}\right)$, where ζ₂₀is the primitive 20-th root of unity then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=8,\left\{1,{\zeta }_{20},{\zeta }_{20}^2,{\zeta }_{20}^3,{\zeta }_{20}^4,{\zeta }_{20}^5,{\zeta }_{20}^6,{\zeta }_{20}^7\right\}$is an integral basis fo 𝕂 and$D_{\mathbb{K}}=2^8·5^6$. If ℳ is a submodule of 𝒪 _𝕂 given by

$$\begin{gathered} \mathcal{M}=\{a_0+a_1{\zeta }_{20}+a_2{\zeta }_{20}^2+a_3{\zeta }_{20}^3+a_4{\zeta }_{20}^4+a_5{\zeta }_{20}^5+a_6{\zeta }_{20}^6+a_7{\zeta }_{20}^7:a_0+a_4\equiv 0(mod4), \\ {a}_1+a_5\equiv 0(mod2),a_2+a_3+a_6\equiv 0(mod4)\text{and}{a}_7\equiv 0(mod5),\text{where} \\ {a}_0,a_1,...,a_7\in \mathrm{\mathbb{Z}}\}, \end{gathered}$$

then $[{𝒪}_{\mathbb{K}}:ℳ]=5$ and

$$\begin{gathered} T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=20(2a_0^2+2a_0{a}_1+5a_0{a}_2+3a_0{a}_3+3a_0{a}_4+2a_0{a}_5+5a_0{a}_6+8a_0{a}_7+a_1^2 \\ +3a_1{a}_2+2a_1{a}_3+2a_1{a}_4+a_1{a}_5+3a_1{a}_6+5a_1{a}_7+4a{2}^2+4a_2{a}_3+5a_2{a}_4 \\ +3a_2{a}_5+7a_2{a}_6+12a_2{a}_7+2a_3^2+3a_3{a}_4+2a_3{a}_5+5a_3{a}_6+7a_3{a}_7+2a_4^2 \\ +2a_4{a}_5+5a_4{a}_6+8a_4{a}_7+a_5^2+3a_5{a}_6+5a_5{a}_7+4a_6^2+12a_6{a}_7+10a_7^2), \end{gathered}$$

where $\alpha \in ℳ$ . Since $t=min\left\{T{r}_{\mathbb{K}}\right(\alpha \overline{\alpha }):\alpha \in ℳ,\alpha \ne 0\}=20$ with $a_1=1$ and $a_0=a_2=a_3=a_4=a_5=a_6=a_7=0$ , it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{20}/2\right)}^8}{2^4·5^4}=\frac{1}{16},$

i.e., the center density of σ(ℳ) is the same of the lattice Λ₈ .

Example 3.7.If$\mathbb{K}=\mathrm{\mathbb{Q}}\left({\zeta }_{21}\right)$, where ζ₂₁is the primitive 21-th root of unity, then$[\mathbb{K}:\mathrm{\mathbb{Q}}]=12,\left\{1,{\zeta }_{21},\cdots ,{\zeta }_{21}^{11}\right\}$is an integral basis of 𝕂 and$D_{\mathbb{K}}=3^6.7^{10}$. If ℳ is a submodule of 𝒪 _𝕂 given by

$$\begin{gathered} \mathcal{M}=\left({\zeta }_{21}^6-{\zeta }_{21}^2+1\right)a_0+\left({\zeta }_{21}^7-{\zeta }_{21}^3+{\zeta }_{21}\right)a_1+\left({\zeta }_{21}^8-{\zeta }_{21}^4+{\zeta }_{21}^2\right)a_2 \\ +\left({\zeta }_{21}^9-{\zeta }_{21}^5+{\zeta }_{21}^3\right)a_3+\left({\zeta }_{21}^{10}-{\zeta }_{21}^6+{\zeta }_{21}^4\right)a_4+\left({\zeta }_{21}^{1}1-{\zeta }_{21}^7+{\zeta }_{21}^5\right)a_5 \\ +\left({\zeta }_{21}^{11}-{\zeta }_{21}^9+{\zeta }_{21}^4-{\zeta }_{21}^3+{\zeta }_{21}-1\right)a_6 \\ +\left({\zeta }_{21}^{11}-{\zeta }_{21}^{10}-{\zeta }_{21}^9+{\zeta }_{21}^8-{\zeta }_{21}^6+{\zeta }_{21}^5-{\zeta }_{21}^3+{\zeta }_{21}^2-1\right)a_7 \\ +\left(-{\zeta }_{21}^{10}+{\zeta }_{21}^8-{\zeta }_{21}^7-1\right)a_8+\left(-{\zeta }_{21}^{11}+{\zeta }_{21}^9-{\zeta }_{21}^8-{\zeta }_{21}\right)a_9 \\ +\left(-{\zeta }_{21}^{11}+{\zeta }_{21}^{10}-{\zeta }_{21}^8+{\zeta }_{21}^6-{\zeta }_{21}^4+{\zeta }_{21}^3-{\zeta }_{21}^2-{\zeta }_{21}+1\right)a_{10} \\ +\left(-{\zeta }_{21}^8+{\zeta }_{21}^7+{\zeta }_{21}^6-{\zeta }_{21}^5-{\zeta }_{21}^2+1\right)a_{11},where{a}_0,a_1,\cdots ,a_{11}\in \mathrm{\mathbb{Z}}\}, \end{gathered}$$

then $[{𝒪}_{\mathbb{K}}:ℳ]=7$ and

$$\begin{gathered} T{r}_{\mathbb{K}}\left(\alpha \overline{\alpha }\right)=28a_0^2-14a_0{a}_2-14a_0{a}_3-14a_0{a}_4+28a_0{a}_5-28a_0{a}_7-14a_0{a}_9 \\ +28a_0{a}_{10}+28a_0{a}_{11}+28a_1^2-14a_1{a}_3-14a_1{a}_4-14a_1{a}_5+28a_1{a}_6 \\ -28a_1{a}_8-14a_1{a}_{10}+28a_1{a}_{11}+28a_2^2-14a_2{a}_4-14a_2{a}_5-14a_2{a}_6 \\ +28a_2{a}_7-28a_2{a}_9-14a_2{a}_{11}+28a_3^2-14a_3{a}_5-14a_3{a}_6-14a_3{a}_7 \\ +28a_3{a}_8-28a_3{a}_{10}+28a_4^2-14a_4{a}_6-14a_4{a}_7-14a_4{a}_8+28a_4{a}_9 \\ -28a_4{a}_{11}+28a_5^2-14a_5{a}_7-14a_5{a}_8-14a_5{a}_9+28a_5{a}_{10}+28a_6^2 \\ -14a_6{a}_8-14a_6{a}_9-14a_6{a}_{10}+28a_6{a}_{11}+28a_7^2-14a_7{a}_9-14a_7{a}_{10} \\ -14a_7{a}_{11}+28a_8^2-14a_8{a}_{10}-14a_8{a}_{11}+28a_9^2-14a_9{a}_{11}+28a_{10}^2 \\ +28a_{11}^2, \end{gathered}$$

where $\alpha \in ℳ$ . Since $t=min\{T{r}_{\mathbb{K}}=(\alpha \overline{\alpha }):\alpha \in ℳ;\alpha \ne 0\}=28$ with $a_0=1$ and $a_1=a_2=\cdots =a_{11}=0$ , it follows that

$\delta \left(\mathcal{M}\right)=\frac{{\left(\sqrt{28}/2\right)}^{12}}{3^3·5^6}=\frac{1}{3^3},$

i.e., the center density of σ(ℳ) is the same of the lattice K₁₂.

ACKNOWLEDGMENT

The authors thank the reviewer for carefully reading the manuscript and for all the suggestions that improved the presentation of this work. The authors also thank FAPESP 2013/25977-7 and CNPq 429346/2018-2 for its financial support.

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Publication Dates

• Publication in this collection
  30 Apr 2020
• Date of issue
  Jan-Apr 2020

History

• Received
  26 Sept 2019
• Accepted
  01 Jan 2020