Hessian matrices, automorphisms of $p$-groups, and torsion points of elliptic curves

We compute the orders of the automorphism groups of finite $p$-groups arising naturally via Hessian determinantal representations of certain elliptic curves defined over number fields. 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 vary with the primes in a "wild", viz. nonquasipolynomial manner.