Let $\eta(\tau)$ be the Dedekind eta function and $f(\tau)=C\frac{\eta(\tau)\eta(4\tau)}{\eta(2\tau)^4}$, with the constant $C=2^{1/4}\eta(i)^2$. My questions concern the minimal polynomials of $f(\frac{i}{2p})$, $p$ an odd prime. The minimal polynomials seem to exhibit simple patterns (see below) and I would like to understand this better. Not being very familiar with algebraic number theory, my questions are: 1) are patterns like these expected? 2) if so, is there an intuitive understanding of them? 3) recommended reading?
Let $M_p(x)$ be the minimal polynomial of $f(\frac{i}{2p})$ when $p\equiv 1 \pmod 4$, and let it be the minimal polynomial of $f(\frac{i}{2p})^4$ when $p\equiv 3 \pmod 4$. A few examples of such minimal polynomials, so we are on the same page:
$$M_{13}(x)=x^3-9x^2+x-1,$$
$$M_{41}(x)=x^{10} - 168222x^9+194609x^8+41520x^7-47346x^6-25332x^5-4530x^4-48x^3+113x^2+18x+1,$$
$$M_{19}(x)=x^5-17041207x^4-236870x^3-6142x^2-11x-\frac{1}{19}.$$
The patterns I am referring to concern the degree 1 coefficient (most surprising to me), the constant term, and the degree of the minimal polynomials.
Degree
The degree of $M_p(x)$ is $\frac{p-1}{4}$ when $p\equiv 1 \pmod 4$ and $\frac{p+1}{4}$ when $p\equiv 3 \pmod 4$.
Constant Term
When $p\equiv 1 \pmod 4$, $M_p(0)=1$ if $p=x^2+32y^2$ for some natural numbers $x$ and $y$, otherwise $M_p(0)=-1$. So, the first $p$ with constant term $1$ are $41, 113, 137, 257, 313, 337, 353$.
When $p\equiv 3 \pmod 4$, $M_p(0)=-\frac{1}{p}$.
Degree 1 Coefficient
For $p\equiv 1\pmod 4$, let $p=even^2+odd^2$, where $even$ and $odd$ are even and odd natural numbers, respectively.
When $p\equiv 5\pmod 8$, the degree 1 coefficient of $M_p(x)$ is $\frac{even}{2}(-1)^{even/4-1/2}$.
When $p\equiv 1\pmod 8$, the degree 1 coefficient of $M_p(x)$ is $$2\left\lfloor \frac{odd}{4}+\frac{1}{4}\right\rfloor(-1)^{\lfloor odd/4-1/4\rfloor}+h(-4p^2)(-1)^{(p+7)/8},$$ where $h(d)$ is the class number for discriminant $d$. So, the degree 1 coefficients for $p=5,13,17,29,37,41,53,61,73,89,97,\ldots,353$ are $1,1,-8,1,-3,18,1,-3,38,42,-44,\ldots,-168$.
I haven't found a simple result yet for the degree 1 coefficient when $p\equiv 3 \pmod 4$, though I haven't looked much. The degree 2 coefficients for $p\equiv 1 \pmod 4$ are very suggestive, but I haven't found the pattern, yet, either.
Are these types of formulas typical for eta quotients? I haven't found many references to general properties of eta quotient minimal polynomials, mainly computations for specific values. Is there any intuition behind any of the details, such as the role of $x^2+32y^2$ in the constant term or the dependence of the degree 1 coefficient on the class number when $p\equiv 1 \pmod 8$, but not when $p \equiv 5 \pmod 8$?
It is important to note that I am not claiming to have proven any of the above, merely to have observed it after finding putative minimal polynomials for $p<200$ and $p\leq 353$ for $p\equiv 1 \pmod 4$ using PARI/GP's algdep function and high precision (e.g., 50,000 digits). (Don't know whether it's worthwhile to share all those minimal polynomials here, but it's straightforward to find them with PARI/GP.)
Thanks!
