There are two definitions of the supersingular primes: one group-theoretic, and the other number-theoretic.
Group-theoretically, let be the modular
group Gamma0, and let be the compactification
(by adding cusps) of , where is the upper half-plane.
Also define
to be the Fricke involution defined by the block matrix . For a prime, define . Then is a supersingular prime if the genus of .
The number-theoretic definition involves supersingular elliptic curves defined over the algebraic closure of the finite
field .
They have their j-invariant in .
Supersingular curves were mentioned by Charlie the math genius in the Season 2 episode "In Plain Sight"
of the television crime drama NUMB3RS.
There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (OEIS A002267). The supersingular
primes are exactly the set of primes that divide the group
order of the Monster group.