  For small odd primes $p$, we prove that most of the rational points on the
modular curve $X_0(p)/w_p$ parametrize pairs of elliptic curves having
infinitely many supersingular primes. This result extends the class of elliptic
curves for which the infinitude of supersingular primes is known. We give
concrete examples illustrating how these techniques can be explicitly used to
construct supersingular primes for such elliptic curves. Finally, we discuss
generalizations to points defined over larger number fields and indicate the
types of obstructions that arise for higher level modular curves.
