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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 $\oplus_{i=1}^{k}{\mathbb{F}_p}^*$

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  • $\begingroup$ What is denoted by $k$? $\endgroup$ – Sasha Aug 30 '16 at 10:09
  • $\begingroup$ k is some integer ! it might be related to the genus of this curve. I just want to say the form of my picard group would be like this ! $\endgroup$ – xyz Aug 30 '16 at 10:20
  • $\begingroup$ Are you looking at the affine curve or its projective completion? The curves you have are rational and the answer to your question can be found in Milnor's book on K-theory, using conductors. $\endgroup$ – Mohan Aug 30 '16 at 13:18
  • $\begingroup$ @Mohan I am looking for Affine solutions. $\endgroup$ – xyz Aug 31 '16 at 5:23
  • $\begingroup$ @Mohan I don't know K-theory, So a bit of details (page number, basic ideas) would be of great help. $\endgroup$ – xyz Aug 31 '16 at 5:48

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