The Hasse invariant of the Tate normal form E5 and the class number of Q(−5l)

Abstract

It is shown that the number of irreducible quartic factors of the form g(x)=x4+ax3+(11a+2)x2−ax+1 which divide the Hasse invariant of the Tate normal form E5 in characteristic l is a simple linear function of the class number h(−5l) of the field Q(−5l), when l≡2,3 modulo 5. A similar result holds for irreducible quadratic factors of g(x), when l≡1,4 modulo 5. This implies a formula for the number of linear factors over Fp of the supersingular polynomial ssp(5⁎)(x) corresponding to the Fricke group Γ0⁎(5).