On the Hasse invariants of the Tate normal forms E5 and E7

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A formula is proved for the number of linear factors over Fl of the Hasse invariant of the Tate normal form E5(b) for a point of order 5, as a polynomial in the parameter b, in terms of the class number of the imaginary quadratic field K=Q(−l), proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term −1, and a theorem is stated for the number of quartic factors of a specific form in terms of the class number of Q(−5l). These results are shown to imply a recent conjecture of Nakaya on the number of linear factors over Fl of the supersingular polynomial ssl(5⁎)(X) corresponding to the Fricke group Γ0⁎(5). The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form E7 for a point of order 7 are determined, which is used to show that the polynomial ssl(N⁎)(X) for the group Γ0⁎(N) has roots in Fl2, for any prime l≠N, when N∈{2,3,5,7}.