Supersingular Prime
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 ![[[0,-1],[N,0]]](/images/equations/SupersingularPrime/Inline6.gif)
.
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.
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basic definition of prime number