We describe the automorphism groups of finite p-groups arising naturally via Hessian determinantal representations of elliptic curves defined over number fields. Moreover, we derive explicit formulas for the orders of these automorphism groups for elliptic curves of j-invariant 1728 given in Weierstrass form. We interpret these orders in terms of the numbers of 3-torsion points (or flex points) of the relevant curves over finite fields. Our work greatly generalizes and conceptualizes previous examples given by du Sautoy and Vaughan-Lee. It explains, in particular, why the orders arising in these examples are polynomial on Frobenius sets and vary with the primes in a nonquasipolynomial manner.

我们描述了通过在数域上定义的椭圆曲线的Hessian行列式表示自然产生的有限p-群的自同构群。此外，我们针对Weierstrass形式给出的j不变式1728的椭圆曲线的这些自同构群的阶数导出了明确的公式。我们根据有限域上相关曲线的3个扭转点（或弯曲点）的数量来解释这些顺序。我们的工作极大地概括和概念化了du Sautoy和Vaughan-Lee给出的先前示例。它特别说明了为什么这些示例中产生的阶数是Frobenius集上的多项式并且以非拟多项式的方式随素数变化的原因。