Supersingular prime (algebraic number theory)

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

Equivalently, for a prime ${\displaystyle p>3}$ of good reduction for ${\displaystyle E}$, the prime is supersingular for ${\displaystyle E}$ if and only if the trace of the Frobenius endomorphism ${\displaystyle a_{p}=p+1-\#E(\mathbb {F} _{p})}$ is zero, that is, ${\displaystyle \#E(\mathbb {F} _{p})=p+1}$. This condition means that the reduction of ${\displaystyle E}$ modulo ${\displaystyle p}$ has the maximum possible endomorphism ring—an order in a quaternion algebra—rather than an order in an imaginary quadratic field.