Let $E$ be the elliptic curve defined over $GF(p)$ and $j$ be j-invariant of $E$, where $p$ is a big prime number. Also suppose $l$ be small prime number (for example $l<5000$) and $\#E$ denote number of points over $E$.



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> Given $j,p$ and $l$, is there deterministic method to find $E$ such that $l$ divide $\#E$?
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Find special elliptic curves from j-invariant
Write down one elliptic curve $E/GF(p)$ with invariant $j(E)=j$. (There's a simple formula to do that.) Use a standard (poly-time) algorithm to compute
$N(E,p):=\#E(GF(p))$. Assuming that $j\ne0$ and $j\ne1728$, then $E$ has a unique quadratic twist over $GF(p)$, so find a non-square in $GF(p)$ and use it to write down an equation for the twist, call it $E'$. Again use the poly-time algorithm, compute $N(E',p):=\#E'(GF(p))$. If either of the numbers $N(E,p)$ or $N(E',p)$ is divisible by $l$, you win. Otherwise there is no such curve over $GF(p)$ with the given $j$-invariant.


However, if you didn't mean to specify the value of $j$, but just want some elliptic curve defined over $GF(p)$ with $l\mid N(E,P)$, then what you need to do is find a point on the modular curve $X\_1(l)$ that is defined over $GF(p)$. 


