Quickly find the cardinality of an elliptic curve

Let $(E:y^2=x^3+ax+b) $ on $\mathbb F_q$, with $ q \mod 2=1$.

If $\gcd(3,q-1)=1$ and $a=0$, then it's easy to find the cardinality of the curve $E$ : $|E|=q$.

Are there an another conditions on $(q, a, b)$, where it's easy to find $|E|$?

The other famous family of this phenomenon is when $b=0$ and $q$ is a prime such that $q\equiv 3\pmod 4$. In this case the number of points is $q+1$. See this theorem whose proof is in Washington's book.

Both are special cases of a wider theory of elliptic curves with complex multiplication (specifically these curves have complex multiplication by $\omega$ and $i$ where $\omega^2+\omega+1=0$ and $i^2+1=0$). CM methods allow us to create curves with a specified number of points and are often used in pairing based cryptography.