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The number of $S_4$ fields with given discriminant

Juergen Klueners
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Abstract

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 $O(p^{1/2}\log(p)^2)$.

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Juergen Klueners. 2005-07-27. The number of $S_4$ fields with given discriminant. https://arxiv.org/abs/math/0411484

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