Let E be an elliptic curve over \({\mathbb{Q}}\), and let p be a prime of supersingular reduction for E. The author shows that the 2-complement of \(E({\mathbb{F}}_ p)\) is cyclic. In particular, if \(E_ a\) is the curve \(y^ 2=(x^ 2+1)(x+a)\) (a\(\in {\mathbb{Q}})\) the author combines the above result with Elkies' theorem (that there are infinitely many primes of supersingular reduction for \(non-C.M.\quad curves)\) to show that there are infinitely many primes p for which \(E_ a({\mathbb{F}}_ p)\) is cyclic. This was previously known assuming the generalized Riemann hypothesis.