  A formula is proved for the number of linear factors and irreducible cubic
factors over $\mathbb{F}_l$ of the Hasse invariant $\hat H_{7,l}(a)$ 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 $\hat H_{7,l}(a)$ 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 $ss_l^{(7*)}(X)$ corresponding
to the Fricke group $\Gamma_0^*(7)$.
