I am currently reading this paper counting $S_3$-sextic fields

- Manjul Bhargava and Melanie Matchett Wood,
*The density of discriminants of $S_3$-sextic number fields*, Proc. Amer. Math. Soc.**136**(2008), 1581-1587. doi:10.1090/S0002-9939-07-09171-X

I'm trying to verify how they obtained the asymptotic constant for $p = 2, 3$.

My attempt:

$$c_2 =\sum_A \mu_2(A) \frac{D(A)}{D(\bar{A})^{1/3}}$$ where the sum $A$ is over all etale cubic $\mathbb{Q}_2$ algebra $A$, $\mu_2(A)$ is the relative density as defined on the paper on page 3 (or 1583), and $\bar{A}$ is the $S_3$ closure of $A$. To compute $\mu_2(A)$, I referred to Davenport-Heilbronn paper

- Manjul Bhargava, Arul Shankar, Jacob Tsimerman,
*On the Davenport-Heilbronn theorems and second order terms*, Invent. math.**193**(2013) 439–499. doi:10.1007/s00222-012-0433-0, arXiv:1005.0672,

in particular Lemma 18 and Lemma 19 in the arXiv version. To look up all the possible extensions, I went over to LMFDB at https://www.lmfdb.org/padicField/.

For example, if $A$ is the cubic extension of $\mathbb{Q}_2$ with ramification index 3 and inertia degree 1, the relative density would be $\frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)}$. The discriminant of $A$, $D(A)$, would be $2^2$ since the discriminant exponent $c=2$, and the discriminant of the $S_3$-closure of $A$ would be $2^4$. I computed this by looking at the Galois splitting model provided in LMFDB, and finding the discriminant of the $p$-part of splitting field.

Doing this for all cubic etale algebra of $\mathbb{Q}_2$, I have that $$c_2 = \frac{\mu(\mathcal{U}_p(111))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(12))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(3))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(1^21))}{\mu(\mathcal{U}_p)}\left(2\frac{p^2}{p^{2/3}} + 4\frac{p^3}{p^{3/3}}\right) + \frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)} \frac{p^2}{p^{4/3}} =\frac{1+2p^{1/3}+4p+p^{-4/3}}{1+p^{-1}+p^{-2}}$$

The 2 and 4 are because there are 2 quadratic extensions of $\mathbb{Q}_2$ with $e=2$ and $c=2$, and 4 quadratic extensions of $\mathbb{Q}_2$ with $e=2$ and $c=3$. I thought maybe I should replace 2 with 2/6, and 4 with 4/6 since there are 6 quadratic extensions with $e=2, f=1$, but it still does not work. Doing the same for $p=3$, I have

$$c_3 = \frac{\mu(\mathcal{U}_p(111))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(12))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(3))}{\mu(\mathcal{U}_p)}\frac{1}{1} + \frac{\mu(\mathcal{U}_p(1^21))}{\mu(\mathcal{U}_p)}\left(2\frac{p}{p^{1/3}}\right) + \frac{\mu(\mathcal{U}_p(1^3))}{\mu(\mathcal{U}_p)}\left(2\frac{p^3}{p^{7/3}} + 3\frac{p^4}{p^{4/3}} + \frac{p^4}{p^{8/3}} + 3\frac{p^5}{p^{11/3}}\right)$$

Additional questions:

Is my computation of the relative density correct? Or have I misunderstood the definition from the paper?

To compute the discriminant of the $S_3$ closure of a cubic extension $A$ of $\mathbb{Q}_p$, is it right to compute the $p$-part of the discriminant of the splitting field of the Galois splitting model of the particular cubic extension $A$?

These are vastly different from what they had on their paper. Obviously, I have a severe misunderstanding/error in my calculation and I would appreciate if someone can point out where my mistakes are in particular.

Thanks in advance!

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