The modular curves $Y(5)$ and $Y_1(5)$ associated to the congruence subgroups $\Gamma(5)$ and $\Gamma_1(5)$ are of genus 0. As such, they are nothing but the Riemann sphere $\mathbb P^1$ minus a finite subset of cusps: $$ Y(5)=\mathbb P^1\setminus C(5) \qquad Y_1(5)=\mathbb P^1\setminus C_1(5)\;. $$ Moreover, it is known that $C(5)$ and $C_1(5)$ are of cardinality 12 and 4 respectively.
Question : what can be taken for the two sets of cusps $C(5)$ and $C_1(5)$?
Once $C(5)$ and $C_1(5)$ are explicitly given, I would like to find an explicit rational expression for the natural quotient map $$ Y(5)\longrightarrow Y_1(5)\,. $$
Of course, I'm aware that this is very classical. A good reference would be welcome.
Thanks.
I'll give you half the answer, and maybe you can work out the other half for yourself.
As I said in a comment a moment ago, there is no canonical isomorphism between $Y_1(5)$ or $Y(5)$ and an open subset of $\mathbf{P}^1$, so $C(5)$ and $C_1(5)$ will depend on your choice of uniformisation. So the real question here is "what are explicit modular functions giving isomorphisms $X_1(5) \cong \mathbf{P}^1$ and $X(5) \cong \mathbf{P}^1$?"
Such a modular function is called a Hauptmodul in the classical theory. For instance, the modular function defined by $$ t(\tau) = q \prod_{n \ge 1} (1 - q^n)^{5 \left(\frac{n}{5}\right)}, $$ where $\left(\frac{n}{5}\right)$ is the Legendre symbol, is a Hauptmodul of level $\Gamma_1(5)$. The four cusps of $\Gamma_1(5)$ are the orbits of $\infty, 2/5, 1/2, 0$ on the boundary of the upper half-plane, and $t(\tau)$ takes these to $0, \infty, (11 \pm 5\sqrt{5})/2$ respectively. (I got these formulae from this paper after a quick web search.) So this gives an explicit isomorphism between $Y_1(5)$ and $\mathbf{P}^1(\mathbf{C}) - \{ 0, \infty, (11 \pm 5\sqrt{5})/2\}$.
Now you can go and look for a Hauptmodul for $\Gamma(5)$ and see if you can repeat this calculation (although it will, of course, be quite messy).