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A formula is proved for the number of linear factors over $\mathbb{F}_l$ ofthe Hasse invariant of the Tate normal form $E_5(b)$ for a point of order $5$,as a polynomial in the parameter $b$, in terms of the class number of theimaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture ofthe author from 2005. A similar theorem is proved for quadratic factors withconstant term $-1$, and a theorem is stated for the number of quartic factorsof a specific form in terms of the class number of $\mathbb{Q}(\sqrt{-5l})$.These results are shown to imply a recent conjecture of Nakaya on the number oflinear factors over $\mathbb{F}_l$ of the supersingular polynomial$ss_l^{(5*)}(X)$ corresponding to the Fricke group $\Gamma_0^*(5)$. The degreesand forms of the irreducible factors of the Hasse invariant of the Tate normalform $E_7$ for a point of order $7$ are determined, which is used to show thatthe polynomial $ss_l^{(N*)}(X)$ for the group $\Gamma_0^*(N)$ has roots in$\mathbb{F}_{l^2}$, for any prime $l \neq N$, when $N \in \{2,3,5,7\}$.

On the Hasse invariants of the Tate normal forms $E_5$ and $E_7$