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In mathematics, there are two notions of supersingular primes.

If is an elliptic curve, then one says that a prime p is supersingular for E if the number of points of E defined over is congruent to 1 modulo p. Equivalently, if the only p-torsion point of E defined over is the identity, then p is a supersingular prime. Note that a supersingular prime need not be singular (in the sense that E modulo p is a singular curve.)

If E has complex multiplication, then the set of primes which are supersingular has Dirichlet density 1/2. However, if E does not have CM, then this density is 0. Nonetheless, Noam Elkies has proven that for any elliptic curve over , there are infinitely many supersingular primes.

An alternative definition of supersingular primes comes from the theory of modular curves. Formally, let H denote the upper half-plane. For a natural number n, let Γ₀(n) denote the modular group Γ₀, 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 ²). As it turns out, there are exactly fifteen supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, and 71 (sequence A002267 in OEIS). It can also be shown that the supersingular primes are exactly the prime factors of the group order of the Monster group M.

Note the set of supersingular primes is a subset of the set of the Chen primes.

• Hasse-Witt matrix

Eric W. Weisstein, Supersingular Prime at MathWorld.

• Joseph H. Silverman (1986). The Arithmetic of Elliptic Curves. Springer.