# r-torsion points on elliptic curve on finite field

Let $E(\mathbb{F}_q)$ - elliptic curve, $r$ is prime, $|E(\mathbb{F}_q)[r]| > 1$. Let $r | q-1$. Is it true that $|E(\mathbb{F}_q)[r]| = r^2$?

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No; let $E$ be $y^2=x^3+x$ over $\mathbf{F}_q$ with $q \equiv 3\pmod{4}$. Then we have $2 \mid q−1$ and $\# E(\mathbf{F}_q)[2] > 1$, but since $−1 \notin \mathbf{F}_q^{\ast 2}$ we have $\# E(\mathbf{F}_q)[2] = 2 < 4$. –  René May 9 '13 at 12:40
@Alex: this might interest you: ipam.ucla.edu/publications/scws1/scws1_6224.pdf –  Dragos Fratila May 9 '13 at 17:14