Can j-invariants be used to decide which elliptic curves are suiteable for cryptography? The j-invariants classify the elliptic curves up to isomorphisms (if we suppose to work in the algebraic closure). Is this classification used in some way to decide whether or not an elliptic curve can be used in cryptography?


If j-invariants can not be used for classification, is there an other classification which responds to my question?
If j-invariants can be used for classification, what does the j-invariants tell us about the curve so that we know it is save to use it for cryptography?


For example: can a certain j-invariant correspond to a curve where we are sure that it has full-torsion so that it is a possible choice for Weil pairing?
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.