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Is it possible to estimate the regulator of a number field of a special form (e.g. cyclotomic fields)?

My motivation mainly comes from the investigation of the number fields $K_n=\mathbb{Q}[x]/(x^n+x+1)$. I have computed their regulators, and they seem to fit an empirical formula

$\text{Reg}(K_n)=(1+O(1))\sqrt{n!} \exp(an)$

for some $a$, where the $(1+O(1))$ term is bounded away from $0$.

EDIT: I have computed the regulators of cyclotomic fields, and the formula above holds as well, with a different $a$ (approximately -1.106).

Question: Is it possible to prove the formula, either for $K_n$ or cyclotomic fields? Or would it be considered hopeless?

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    $\begingroup$ 1. If you don't know the constant $a$, could you really have enough numerical evidence that the factor in front is $(1+O(1))$ and not a weaker subexponential term? 2. You could try to prove this by estimating every other term appearing in the class number formula. The discriminant will match $n!$ up to an exponential factor, and $r_1,r_2$ and $w_K$ should be easy to compute, so that leaves the class number and the residue. The class number, you note, is $1$ - this should be very hard to prove but one can give a heuristic. $\endgroup$
    – Will Sawin
    Commented Apr 22, 2020 at 13:59
  • $\begingroup$ It might be possible to prove upper and lower bounds on the residue, maybe even subexponential ones (though I doubt it). 3. The way you wrote your formula, it only implies an upper bound and not a lower bound, do you really mean this? $\endgroup$
    – Will Sawin
    Commented Apr 22, 2020 at 14:03
  • $\begingroup$ 1. In all cases I've computed, the $(1+O(1))$ is between 0.2 and 0.6. $\;\;\;$ 2. The purpose to compute the regulator is to prove that the class number is $1$, which I believe there's no way to compute other than the Class Number Formula, so I would be happy if there're direct methods (or even heuristics) for determining class numbers. $\endgroup$
    – LeechLattice
    Commented Apr 22, 2020 at 14:05
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    $\begingroup$ The Cohen-Lenstra heuristics predict that the average class number of a "random" field with $k$ places at $\infty$ is something like $\zeta(k)$, so for $k$ large they predict the class number is $1$ with high probability. I can dig out a precise reference... $\endgroup$
    – Will Sawin
    Commented Apr 22, 2020 at 14:12

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