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Using the theory of elliptic curves, we show that the class number hðÀpÞ of the field Qð ffiffiffiffiffiffi ffi Àp p Þ appears in the count of certain factors of the Legendre polynomials P m ðxÞ ðmod pÞ; where p is a prime 43 and m has the form ðp À eÞ=k; with k ¼ 2; 3 or 4 and p e ðmod kÞ: As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2 þ axy þ y ¼ x 3 and find an elementary expression for the supersingular polynomial ss p ðxÞ whose roots are the supersingular j-invariants of elliptic curves in characteristic p: As a corollary we show that the class number hðÀpÞ also shows up in the factorization ðmod pÞ of certain Jacobi polynomials.

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Abstract.An improvement is given to the method of Hermite for finding a and b in p = a1 + b2, where p is a prime m 1 (mod 4).