The Hasse invariant of the Tate normal form $E_7$ and the supersingular polynomial for the Fricke group ${\mathrm{\Gamma }}_0^{\ast }(7)$

A formula is proved for the number of linear factors and irreducible cubic factors over ${\mathbb{F}}_l$ of the Hasse invariant ${\widehat{H}}_{7,l}(a)$ of the Tate normal form $E_7(a)$ for a point of order $7$, as a polynomial in the parameter $a$, in terms of the class number of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$. Conjectural formulas are stated for the numbers of quadratic and sextic factors of ${\widehat{H}}_{7,l}(a)$ of certain specific forms in terms of the class number of $\mathbb{Q}(\sqrt{-7l})$, which are shown to imply a recent conjecture of Nakaya on the number of linear factors over ${\mathbb{F}}_l$ of the supersingular polynomial $s{s}_l^{(7\ast )}(X)$ corresponding to the Fricke group ${\mathrm{\Gamma }}_0^{\ast }(7)$.