  We prove that the number of quartic $S_4$--extensions of the rationals of
given discriminant $d$ is $O_\eps(d^{1/2+\eps})$ for all $\eps>0$. For a prime
number $p$ we derive that the dimension of the space of octahedral modular
forms of weight 1 and conductor $p$ or $p^2$ is bounded above by
$O(p^{1/2}\log(p)^2)$.
