I was trying to calculate the number of points of the curve $E:y^2 = x^3 + 4x^2 + 2x$ over $\mathbb{F}_p$ for $p\equiv 1\bmod 8$ (In order to have $\sqrt{-2}\in\mathbb{F}_p$) but I did not succeed. This curve is mentioned in Silverman's Advanced Topics in the Arithmetic of Elliptic curves (Proposition 2.3.1) to have multiplication by $\sqrt{-2}$.

Over these primes $p\equiv 1\bmod 8$ the curve $E$ has full $2$-torsion so $E(\mathbb{F}_p)\cong \mathbb{Z}/(2)\times \mathbb{Z}/(k)$.

I would like to have an elliptic curve with CM by $\sqrt{-2}$ such that I can know the number of points in terms of $p$. Does anybody has a suggestion? or maybe another curve?

Thanks