Supersingular prime

In mathematics, a supersingular prime is a certain kind of prime number.

Formally, let H denote the upper half plane. For a natural number n, let Γ₀(n) denote the modular group Gamma0, and let w_n be the Fricke involution defined by the block matrix [[0, -1], [n, 0]]. Furthermore, let the modular curve X₀(n) be the compactification (with added cusps) of

Y₀(n) = Γ₀(n)\H,

and for any prime p, define

X₀⁺(p) = X₀(p) / w_p.

Then p is supersingular means by definition that the genus of X₀⁺(p) is zero.

It is also possible to define supersingular primes in a number-theoretic way using supersingular elliptic curves defined over the algebraic closure of the finite field GF(p) that have their j-invariant in GF(p). [details, anyone?]

As is turns out, there are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 . It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.