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