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It is shown that the number of irreducible quartic factors of the form $g(x)= x^4+ax^3+(11a+2)x^2-ax+1$ which divide the Hasse invariant of the Tate normalform $E_5$ in characteristic $l$ is a simple linear function of the classnumber $h(-5l)$ of the field $\mathbb{Q}(\sqrt{-5l})$, when $l \equiv 2,3$modulo $5$. A similar result holds for irreducible quadratic factors of $g(x)$,when $l \equiv 1, 4$ modulo $5$. This implies a formula for the number oflinear factors over $\mathbb{F}_p$ of the supersingular polynomial$ss_p^{(5*)}(x)$ corresponding to the Fricke group $\Gamma_0^*(5)$.

The Hasse invariant of the Tate normal form $E_5$ and the class number of $\mathbb{Q}(\sqrt{-5l})$