Immediate Descendants of the Elliptic-Curve Groups of Maglione and Stanojkovski

For a prime \(p\ge 5\), an elliptic curve \(E\colon y^2=x^3+ax+b\) over \(\mathbb{F}_p\) and a point \(P\in E(\mathbb{F}_p)\) other than the origin, Maglione and Stanojkovski defined a group \(G_{E,P}(\mathbb{F}_p)\) of order \(p^9\), class \(2\) and exponent \(p\). In the problem session of Oberwolfach Report 27/2025, Maglione asked which pairs \((E,P)\) yield an immediate descendant, defined there as a central extension by \(\mathbb{Z}/p\mathbb{Z}\) of class \(3\) and order \(p^{10}\), and remarked that it would be remarkable if the sum, over the isomorphism classes of these groups, of the numbers of immediate descendants were a PORC function of \(p\). We prove that \(G_{E,P}(\mathbb{F}_p)\) has an immediate descendant if and only if \(P\) has order \(2\), and is terminal otherwise; for \(P\) of order \(2\) the nucleus has rank \(2\). For these pairs we give the number of immediate descendants of order \(p^{10}\), and of those of exponent \(p\), as explicit functions of \(p\), of \(|E[3](\mathbb{F}_p)|\) and of the order of the stabilizer of \(P\) in the automorphism group of \(E\); this contains the theorem of du Sautoy and Vaughan-Lee for \(y^2=x^3-x\) and \(P=(0,0)\). Summed over the isomorphism classes, both numbers are PORC, with moduli \(72\) and \(12\): the dependence on \(|E[3](\mathbb{F}_p)|\) cancels because the pairs \((E,P)\) with given \(3\)-torsion are parametrized by curves of genus \(0\). This answers both questions with the source's definition of an immediate descendant. If the descendants of order \(p^{11}\), which are immediate descendants in O'Brien's sense, are counted as well, the PORC question remains open. The question is Maglione's; the groups, their classification and their automorphism groups are due to Maglione and Stanojkovski. This is an unrefereed note.