Supersingular prime
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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 wn 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) / wp.
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 Template:OEIS. It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.Template:Math-stub