@MingcongZeng: This is Morava's change of rings theorem. Map $S$ to $L_{K(n)}S^0$, and look at the induced map. in $\mathrm{Ext}_{BP_*}$, and then use the change of rings theorem.

@KonradVoelkel - I've added what I believe to be the connection, hopefully it is correct!

Perhaps this is what you mean. Take $E = K(1)$, the first Morava $K$-theory. You can find the homotopy groups $\pi_*L_{K(1)}S^0$, at least when $p$ is odd, in, for example, Lurie's course notes (math.harvard.edu/~lurie/252xnotes/Lecture35.pdf). The order of the cyclic summand that appears can be expressed as the denominator of a certain expression involving Bernoulli numbers. This is related to the image of $J$, see en.wikipedia.org/wiki/J-homomorphism

I've kept it open in the hope that someone comes along and tells me that the sketch I outlined in the question is correct! I'm still working through the answer you gave; it has quite a few references to sort through. There is a topological nature to the question, which unfortunately I cannot adequately describe in the comments.

@FernandoMuro: Yes, the sequence $x_0,\ldots,x_{n-1}$ is assumed to be regular