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We provide a direct computation of the $F$-pure threshold of degree fourhomogeneous polynomial in two variables and, more generally, of certainhomogeneous polynomials with four distinct roots. The computation depends onwhether the cross ratio of the roots satisfies a specific M\"{o}biustransformation of a Legendre polynomial. We then make a connection between along lasting open question, involving the relationship between the $F$-pure andthe log canonical threshold, and roots of Legendre polynomials over$\mathbb{F}_p$.

Legendre Polynomials Roots and the $F$-Pure Threshold of bivariate Forms