$j$-invariants for elliptic curves over $\mathbb{F}_p$ I'm reading an article about elliptic curve volcanos. I know how to compute the $j$-invariant given a curve in Weierstrass form, but i don't have any idea on how to compute every possible $j$-invariant possible for curves defined over $\mathbb{F}_p$, other than brute forcing every Weierstrass form curve.
In the paper the number of $j$-invariants is finite and every one of them is smaller than $p$. 

How were those computed?
 A: A curve defined over a given field $K$ the $j$-invariant of an elliptic curve is an element of that field. Therefore for a finite field of prime order the $j$-invariant can be represented by a number less than $p$.
As for which $j$-invariants are possible, they all are! The curve
$$ y^2 + xy = x^3 - \frac{36}{j_0 - 1728} x - \frac{1}{j_0 - 1728} $$
is well known and you can calculate its $j$-invariant to be $j_0$, the only edge case is $j_0 = 1728$ where the above formula breaks down, nevertheless, an elliptic curve of $j$-invariant 1728 are given by $y^2 = x^3 - x$ for $p\ne 2$.
In particular I assume that in the diagram it is not implied that those are all cordilliera, just some examples of some.
A: You may also want to look into modular polynomials. These are polynomials $\Phi_\ell(X, Y) \in \mathbb{Z}[X,Y]$ for primes $\ell$ such that the roots of $\Phi_\ell(X, j_0)$ over $\mathbb{F}_p$ are the $j$-invariants of curves over $\mathbb{F}_p$ that are $\ell$-isogenous to the curve $j_0$. The construction of these polynomials is complicated, but you can access them in Sage (from polmodular in Pari/GP). The degree of $\Phi_\ell(X, j_0)$ is $\ell + 1$ so, depending on the size of $p$, it can be much faster to just factor $\Phi_\ell(X, j_0)$ over $\mathbb{F}_p$ than to enumerate all the $j$-invariants and check them individually. You can use Vélu's formulae to write down the explicit isogeny if you want to see that (available as ellisogeny in Pari/GP, not sure for Sage).
