The generalized Weierstrass curve-count conjecture over integer residue rings

Let $n$ be a positive integer, and let $N_G(\mathbb{Z}_n)$ denote the number of nonsingular generalized Weierstrass elliptic curves over $\mathbb{Z}_n$. Define the discriminant quantities by

$$\Delta=-b_2^2b_8-8b_4^3-27b_6^2+9b_2b_4b_6,$$

with

$$b_2=a_1^2+4a_2,\qquad b_4=2a_4+a_1a_3,$$ $$b_6=a_3^2+4a_6,\qquad b_8=a_1^2a_6+4a_2a_6-a_1a_3a_4+a_2a_3^2-a_4^2.$$

Generalized Weierstrass counting conjecture.

$$N_G(\mathbb{Z}_n)=\Phi(n^5).$$

The formula is supported by computational validation in the paper; no resolution status is supplied.