I'm working with the modular curves $X_0(p)$ for $p=2,5$, with
$$f_2=q\prod_{n\geq1}(1+q^n)^{24}$$ and $$f_5=q\prod_{n\geq1}(1+q^n+q^{2n}+q^{3n}+q^{4n})^6$$ being inverses of the Hauptmodul and uniformisers of $X_0(2)$ and $X_0(5)$ respectively.
I'm trying to find the elliptic points on $X_0(p)$ and the values of $f_p$ at these points. I've been told that by writing a modular relation $P(f)/f = j$ where $P$ is a polynomial and $j$ the $j$-function, such points correspond to $j=0$ or $1728$ and the map $P(f)/f$ being unramified.
I'm happy with $f_2=-2^{-6}$ corresponding to $j=1728$ once one writes $(1+256f_2)^3/f_2 = j$, but how do I check if the map above is unramified?
In the $p=5$ case, this paper seems to suggest that $f_5$ takes the value of $\pm 5^{-3/2}$ at an elliptic point of $X_0(5)$ (equation 3.19, comparing with the various inequalities from section 3.1), but I can't seem to use the modular relation at all if this is the case. I get
$$\frac{(1+250f_5+3125f_5^2)^3}{f_5} = j$$
but then $f_5=\pm5^{-3/2}$ doesn't correspond at all to $j=0$ or $1728$, and indeed such values of $f$ are rather horrible... If anyone could shed any light on finding these points and the values of $f$ at them, it would be much appreciated.
EDIT: I've since discovered that the map is ramified at a point if and only if its derivative vanishes there. Using this, it's easy to determine that $-2^{-6}$ is the only $f_2$ value which gives $P(f_2)/f_2 = 0$ or $1728$ with non-vanishing derivative.
As for $p=5$, the only values of $f_5$ for which $(1+250f_5+3125f_5^2)^3/f_f = 0$ or $1728$ and the derivative is nonzero are $f_5 = 5^{-3}(-11\pm 2i)$, which has norm $5^{-3/2}$, as (sort of) conjectured.
I've since discovered that the map is ramified at a point if and only if its derivative vanishes there. Using this, it's easy to determine that $-2^{-6}$ is the only $f_2$ value which gives $P(f_2)/f_2 = 0$ or $1728$ with non-vanishing derivative.
As for $p=5$, the only values of $f_5$ for which $(1+250f_5+3125f_5^2)^3/f_f = 0$ or $1728$ and the derivative is nonzero are $f_5 = 5^{-3}(-11\pm 2i)$, which has norm $5^{-3/2}$, as (sort of) conjectured.