In 1945–1946, C. L. Siegel proved that an $n$-dimensional lattice $\unicode[STIX]{x1D6EC}$ of determinant $\text{det}(\unicode[STIX]{x1D6EC})$ has at most $m^{n^{2}}$ different sublattices of determinant $m\cdot \text{det}(\unicode[STIX]{x1D6EC})$. In 1997, the exact number of the different sublattices of index $m$ was determined by Baake. We present a systematic treatment for counting the sublattices and derive a formula for the number of the sublattice classes under unimodular equivalence.

中文翻译：

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格子的分类

1945-1946 年，CL Siegel 证明了$n$维晶格$\unicode[STIX]{x1D6EC}$行列式的$\text{det}(\unicode[STIX]{x1D6EC})$最多有$m^{n^{2}}$行列式的不同亚格$m\cdot \text{det}(\unicode[STIX]{x1D6EC})$. 1997年，指数不同亚格的确切数目$m$由巴克决定。我们提出了一种计算子格的系统处理方法，并推导出了单模等价下子格类数的公式。