In algebraic number theory, a **supersingular prime** is a prime number with a certain relationship to a given elliptic curve. If the curve *E* defined over the rational numbers, then a prime *p* is **supersingular for ***E* if the reduction of *E* modulo *p* is a supersingular elliptic curve over the residue field **F**_{p}.

Elkies (1987) showed that any elliptic curve over the rational numbers has infinitely many supersingular primes. However, the set of supersingular primes has asymptotic density zero. Lang & Trotter (1976) conjectured that the number of supersingular primes less than a bound *X* is within a constant multiple of
X
ln
X
, using heuristics involving the distribution of eigenvalues of the Frobenius endomorphism. As of 2012, this conjecture is open.

More generally, if *K* is any global field—i.e., a finite extension either of **Q** or of **F**_{p}(*t*)—and *A* is an abelian variety defined over *K*, then a **supersingular prime
p
for ***A* is a finite place of *K* such that the reduction of *A* modulo
p
is a supersingular abelian variety.