The Klein quartic as a cyclic group gene

Let $k$ be a field, and let $a$, $b$, $c$ be three elements in $k^X$. The nonsingular projective plane curve $X$ defined over $k$ with equation
$$ ax^3y+by^3z+cz^3x=0 $$
has genus 3 and is reduced to the familiar Klein quartic when $a=b=c=1$. The Jacobian $J_X$ of $X$ is a three-dimensional abelian variety, defined over $k$ as well. The aim of this article is to give some formulas for the number of points of the group $J_X(k)$ of rational points of $J_X$ if $k=\mathbb F_q$ is a finite field.
We assume that the full group of seventh roots of unity is contained in $k$; this amounts to saying that $q\equiv 1$ (mod 7). If q is a prime number and the coefficients $a$, $b$, $c$ are appropriately chosen, we noticed that the number of points of the group $J_X(k)$ is prime in a significant number of occurences. This provides cyclic groups which seem to be accurate for cryptographic applications.