Let $r\ge 3$ be an odd integer and ${\mathbb{F}}_{q^{2r}}$ the finite field with $q^{2r}$ elements. A second generalisation of the Giulietti-Korchmáros maximal curve over ${\mathbb{F}}_{q^6}$ was presented in 2018 by Beelen and Montanucci, the so-called BM curve. This curve is maximal over ${\mathbb{F}}_{q^{2r}}$ and isomorphic to the Giulietti-Korchmáros curve for $r=3$. In this paper, benefiting from suitable representations of the automorphism group of the BM curve, we construct explicit equations for families of maximal algebraic curves as Galois subcovers of the BM curve, we also provide the genus and the Galois group associated to the subcover.

中文翻译：

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最大曲线的显式方程作为 BM 曲线的子覆盖

让 $r\ge 3$ 是一个奇数并且 ${\mathbb{F}}_{q^{2r}}$ 有限域 $q^{2r}$元素。Giulietti-Korchmáros 极大曲线的第二个推广${\mathbb{F}}_{q^6}$由 Beelen 和 Montanucci 于 2018 年提出，即所谓的BM曲线。这条曲线最大${\mathbb{F}}_{q^{2r}}$ 与 Giulietti-Korchmáros 曲线同构 $r=3$. 在本文中，受益于BM曲线的自同构群的适当表示，我们构造了最大代数曲线族的显式方程作为BM曲线的Galois子覆盖，我们还提供了与子覆盖相关的属和Galois群。