Can j-invariants be used to decide which elliptic curves are suiteable for cryptography?
Giving the j-invariant is basically equivalent to giving a curve equation (up to quadratic twist) (over $\mathbb F_p$ for $p &gt; 3$) : to each j-invariant you can attach a pair of curves $(E, \widetilde E)$. You can easily compute the curve equations of $E$ and $\widetilde E$ from $j$: 

$$E: y^2 = x^3 + a x + 2a, \qquad a = -\frac{27j}{j-1728}$$

Now the $j$-invariant is "suitable for crypto" if both curves above are suitable, which you can check using the usual algorithms (SEA, etc.).
Since it is easy to go from “curve equation” to “j-invariant” and back, it is also easy to conclude that the j-invariant does not contain anything particularly useful for cryptography.