Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory of elliptic curves, we show that the class number h(−p) of the field appears in the count of certain factors of the Legendre polynomials Pm(x) (mod p), where p is a prime >3 and m has the form (p−e)/k, with k=2,3 or 4 and p≡e (mod k). As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y2+αxy+y=x3 and find an elementary expression for the supersingular polynomial ssp(x) whose roots are the supersingular j-invariants of elliptic curves in characteristic p. As a corollary we show that the class number h(−p) also shows up in the factorization (mod p) of certain Jacobi polynomials.