The modular covers $X_0(2)\to X(1)$ and $X_0(6)/w_2\to X_0(2)/w_2$ as Belyi maps

Define a rational coordinate $h_2$ on the modular curve $X_0(2)$ by $$ h_2(z) = \bigl(\eta(z) \, / \, \eta(2z)\bigr)^{24} = q^{-1} \prod_{n=1}^\infty \, (1+q^n)^{-24} = q^{-1} - 24 + 276 q - 2048 q^2 + 11202 q^3 - 49152 q^4 + - \cdots. $$ Then $j = (h_2 + 256)^3 / h_2$. As expected, this degree-3 map is ramified only above $j=0$ (with multiplicity 3) and $j=\infty,1728$ (each with multiplicities $2+1.)$ To put this function in our standard form, we apply fractional linear transformations to $j$ and $h_2$ to put $j=0$ and $h_2 = -256$ at infinity; one choice, which also puts the preimages $0,\infty$ of $j=\infty$ at $0,1$ is $$ j = 1728 \, / \, t, \quad h_2 = 256 \, x \, / \, (1-x) $$ and this recovers our formula $t = \frac{3^3}{1^1 2^2} x^2 (1-x)$ for a degree-3 Belyi polynomial.

The modular curve $X_0(2)$ itself has only three critical points, with coordinates $h_2 = \infty$ and $h_2 = 0$ (the cusps $z=i\infty$ and $z=0$) and $h_2 = -64$ (the simple preimage of $j=1728,$ corresponding to the elliptic point $z = (1+i)/2).$ The Fricke involution $w_2$ on $X_0(2)$ takes $h_2$ to $2^{12} / h_2$ [exercise: derive this formula from the identities $w_2(z) = -1/2z$ and $\eta(-1/z) = (z/i)^{1/2} \eta(z)$]. Thus the quotient curve $X_0^+(2) = X_0(2) / w_2$ has rational coordinate $$ h_2^+(z) = (h_2(z) + 64)^2 \, / \, h_2(z) = q^{-1} + 104 + 4372 q + 96256 q^2 + 1240002 q^3 + \cdots, $$ and three critical points $h_2 = \infty, 0, 256$ of index $\infty,4,2$ corresponding to $z = i\infty, (1+i)/2, i/\sqrt 2$. [NB the last of these corresponds to $h_2 = +64,$ which indeed maps to $j = 20^3 = j(\sqrt{-2}) = j(i/\sqrt 2).]$

We next compute the degree-4 map $X_0(6)/w_2 \to X_0(2)/w_2 = X_0^+(2).$ The curve $X_0(6)/w_2$ has rational coordinate $$ h_6^+ = \left( \frac{\eta(z) \, \eta(2z)}{\eta(3z) \, \eta(6z)} \right)^4 = q^{-1} - 4 - 2 q + 28 q^2 - 27 q^3 - 52 q^4 + 136 q^5 - 108 q^6 - 162 q^7 \cdots. $$ We compare $q$-expansions to write $h_2^+$ as a rational function of degree $4$ in this $h_6^+$. In gp this is done conveniently with the code 


h2 = (eta(q) / eta(q^2))^24 / q

h6 = ( (eta(q) * eta(q^2)) / (eta(q^3) * eta(q^6)) )^4 / q

H2 = subst( (h2+64)^2/h2, q, serreverse(1/h6) )

subst(truncate(H2), q, 1/h)
[The gp function eta takes a power series $X$ to $\prod_{n=1}^\infty (1-X^n)$ without the $X^{1/24}$ (there is no support for fractional power series in gp) so we had to supply the “/q” ourselves.] The last step works because the only poles of $h_2^+$ on $X_0(6)/w_2$ are at $h_6^+ = 0, \infty,$ so the power series H2, which gives $h_2^+$ in terms of $1/h_6^+$, is simply $q^{-1} + 108 + 4374 q + 78732 q^2 + 531441 q^3 + O(q^{16}).$ Factoring the final result, we obtain $$ h_2^+ = (h_6^+ + 27)^4 \, / \, {h_6^+}^3 $$ with $h_2^+ - 256 = (h_6^+ - 81)^2 \, ({h_6^+}^2 + 14 h_6^+ + 81) \, / \, {h_6^+}^3.$ We conclude as before with a change of coordinates, here $$ h_2^+ = 256 \, / \, t, \quad h_6^+ = 27 \, x \, / \, (1-x), $$ to recover our formula $t = \frac{4^4}{1^1 3^3} x^3 (1-x)$ for a degree-4 Belyi polynomial.
In general the availability of $q$-expansions of rational functions on modular curves makes it feasible to compute formulas for these curves and for maps between them whose degree is much larger than would be feasible for a “random” Belyi cover. We shall do such computations later in the course.

The modularity of $h_2$ and $h_6^+$ follows from Ligozat’s criterion(*) for the modularity of an eta product. They have degree $1$ on $X_0(2)$ and $X_0(6)/w_2$ respectively because an eta product has zeros and poles only at cusps, and in each case there is just one simple pole (at $z=i\infty$) and one simple zero.

(*) G. Ligozat: Courbes modulaires de genre 1, Mém. Soc. Math. de France 43 (1975) = eudml.org/doc/94716. See Proposition 3.2.1, starting on page 28.
We use the letter $h$ in $h_2, h_2^+, h_6^+$ to suggest “Hauptmodul”. In general, a modular curve $X$ of genus $0$ with a cusp of width $k$ at infinity always has a rational coordinate of the form $q^{-1/k} + O(1),$ defined uniquely up to an additive constant; such a function is called a “Hauptmodul” on $X$. For example,  $j$ is a Hauptmodul on $X(1).$ The plural of “Hauptmodul” is “Hauptmoduln”.