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A formula is proved for the number of linear factors and irreducible cubicfactors over $\mathbb{F}_l$ of the Hasse invariant $\hat H_{7,l}(a)$ of theTate normal form $E_7(a)$ for a point of order $7$, as a polynomial in theparameter $a$, in terms of the class number of the imaginary quadratic field$K=\mathbb{Q}(\sqrt{-l})$. Conjectural formulas are stated for the numbers ofquadratic and sextic factors of $\hat H_{7,l}(a)$ of certain specific forms interms of the class number of $\mathbb{Q}(\sqrt{-7l})$, which are shown to implya recent conjecture of Nakaya on the number of linear factors over$\mathbb{F}_l$ of the supersingular polynomial $ss_l^{(7*)}(X)$ correspondingto the Fricke group $\Gamma_0^*(7)$.

The Hasse invariant of the Tate normal form $E_7$ and the supersingular polynomial for the Fricke group $\Gamma_0^*(7)$