= Solution
If $p\mid D$, square-freeness gives the cuspidal reduction $y^2=x^3$. Its nonsingular group is isomorphic to the <additive group> $\mathbb F_p$, so it is cyclic of order $p$.
Suppose $p\nmid D$. The reduction is an <elliptic curve>. Since $p\equiv3\pmod4$, $-1$ is a <quadratic nonresidue>. Pairing $x$ with $-x$ in the quadratic-character sum and using
$$
(-x)^3-D^2(-x)=-(x^3-D^2x)
$$
shows that the two contributions cancel. Therefore $\#\widetilde E(\mathbb F_p)=p+1$. The three roots $0,D,-D$ are distinct, so the full <2-torsion> is rational. A cyclic group has at most two elements killed by $2$, hence $\widetilde E(\mathbb F_p)$ is noncyclic. Thus
$$
\boxed{\widetilde E_{\mathrm{ns}}(\mathbb F_p)\text{ is cyclic of order }p\text{ if }p\mid D,\text{ and noncyclic of order }p+1\text{ otherwise}.}
$$
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