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

## TL;DR

This paper derives formulas for the factorization of Hasse invariants of Tate normal forms $E_5$ and $E_7$, linking their factors to class numbers of quadratic fields, and confirms related conjectures about supersingular polynomials.

## Contribution

It provides explicit formulas for the number of linear and quadratic factors of Hasse invariants in terms of class numbers, and determines the roots of supersingular polynomials for specific modular groups.

## Key findings

- Formulas for linear factors of Hasse invariants in terms of class numbers.
- Determination of irreducible factors of the Hasse invariant for $E_7$.
- Roots of supersingular polynomials lie in quadratic extensions for certain groups.

## Abstract

A formula is proved for the number of linear factors over $\mathbb{F}_l$ of the 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 the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-l})$, proving a conjecture of the author from 2005. A similar theorem is proved for quadratic factors with constant term $-1$, and a theorem is stated for the number of quartic factors of 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 of linear factors over $\mathbb{F}_l$ of the supersingular polynomial $ss_l^{(5*)}(X)$ corresponding to the Fricke group $\Gamma_0^*(5)$. The degrees and forms of the irreducible factors of the Hasse invariant of the Tate normal form $E_7$ for a point of order $7$ are determined, which is used to show that the 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 $N \in \{2,3,5,7\}$.