In Washington’s “Elliptic curves number theory and cryptography” section 4.4, he mentions a special family of curves. This family is of the form $y^2=x^3-kx$ over a prime field $F_p$. The interesting part about this family is that the curve’s order is given by a formula that depends on $p$ and $k$.
What are other examples of such interesting families? Where the curve order is given by an explicit formula depending on the family’s parameters?
(And I am aware of cases that produce only the order $p+1$. So I’m curious for other, more interesting cases)