# Do the elliptic curves over prime fields must always contain prime number of elements (prime order)?

I have gone through one example where i saw a curve defined over some prime number containing non-prime order.

• You've just seen an example of a curve over a prime field which has non-prime order, and your question is whether they exist? How could you have found one if they don't? – CurveEnthusiast Jan 25 '17 at 8:01
• Don't you think you just answered your own question? – fkraiem Jan 25 '17 at 11:00
• Montgomery and Edwards curves can't have a prime order. – CodesInChaos Jan 25 '17 at 13:51

In general, an elliptic curve over $\mathbb{F}_p$ can have a nonprime order. Much more is known.
As mentioned in Stinson's book Cryptography:Theory and Practice, 2nd Edition, Theorem 6.1, the additive group of an Elliptic curve $E$ over $\mathbb{F}_p$ with $p>3,$ and prime, is isomorphic to $Z_{n_1}\times Z_{n_2}$ with $$n_2|n_1,\quad n_2|(p-1).$$