Consider an Elliptic curve with a singularity (double point) over $\mathbb{F}_p$ $$y^2=x^2(x-a) $$ where $a$ is a quadratic residue in ${\mathbb{F}_p}^*$.
We can easily count the number of (non-singular)points on this curve over ${\mathbb{F}_p}$ (They are $p-1$). Also it is well known that $$E_{ns}({\mathbb{F}_p}) \cong {\mathbb{F}_p}^* $$
Now consider an exact analogous condition, Now my curve is $$ y^3=x^3(x-a) $$ over ${\mathbb{F}_p}$ such that $3|p-1$ . The nonsingular points on this curve is again $p-1$.
My question is does the group associated with this curve (Picard group or Jacobian variety) known ? My guess would be $\oplus_{i=1}^{k}{\mathbb{F}_p}^*$