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.
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 ![[[0,-1],[N,0]]](/images/equations/SupersingularPrime/Inline6.svg)
. For 
a prime, define 
. Then 
is a supersingular prime if the genus of 
.
.
They have their j-invariant in 
.
