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

For a prime p ≥ 5, an elliptic curve E: y² = x³ + ax + b over F_p and a point P ∈ E(F_p) other than the origin, Maglione and Stanojkovski defined a group G_{E,P}(F_p) of order p⁹, 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 Z/pZ of class 3 and order p¹⁰, 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}(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 p¹⁰, and of those of exponent p, as explicit functions of p, of |E[3](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 y² = x³ − x and P = (0,0). Summed over the isomorphism classes, both numbers are PORC, with moduli 72 and 12: the dependence on |E[3](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 p¹¹, 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. Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text. Corpus identifier: OWR-14299292-015 (J. Maglione, problem session, Oberwolfach Report 27/2025, pp. 1443-1444).