2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/73322We 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 $O(p^{1/2}\log(p)^2)$.new versionNumber Theory11R29;11R16, 11R32text