Abstract
The factorization of the Legendre polynomial of degree ( p − e ) / 4 , where p is an odd prime, is studied over the finite field F p . It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism − 2 p , by proving an analogue of Deuring's theorem on supersingular curves with multiplier − p . This is used to count the number of irreducible binomial quadratic factors of P ( p − e ) / 4 ( x ) over F p in terms of the class number h ( − 2 p ) .
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