I know (edit: three) families of smooth projective connected curves over $\bar{\mathbf{Q}}$ for which the Belyi degree is not hard to bound from above.

The modular curves $X(n)$. They are constructed by compactifying the quotient $Y(n) = \Gamma(n)\backslash \mathbf{H}$. The natural morphism $X(n) \longrightarrow X(1)$ is Belyi of degree $n^2$ (up to a constant factor). This also bounds the Belyi degree of a modular curve given by a congruence subgroup $\Gamma$. In general, Zograf proves that the Belyi degree of a (classical congruence) modular curve is bounded by $128(g+1)$.

The Fermat curves $F(n)$. They are given by the equation $x^n+y^n+z^n =0$ in $\mathbf{P}^2$. The morphism $(x:y:z)\mapsto (x^n:z^n)$ is Belyi of degree $n^2$. It is known that $F(n)$ is

**not**a modular curve for $n$ big enough. So this example is really different than the one above. (Also note that $n^2\leq 10g+10$ by the Plucker formula.)Wolfart curves are curves $X$ over $\overline{\mathbf{Q}}$ with a Galois Belyi morphism $X\to \mathbf{P}^1$; I took this terminology from a preprint by Pete L. Clark. Such curves are also called Galois Belyi covers or Galois three-point covers in the literature. The Belyi degree of a Wolfart curve is bounded by $84(g-1)$. (In particular, the latter implies that there are only finitely many Wolfart curves of given genus.)

The following family of curves is not so easily dealt with.

- For an elliptic curve $E$ over the rational numbers, the Belyi degree can be bounded in the height of the $j$-invariant of $E$ following Belyi's proof of his theorem. This was written down explicitly by Khadjavi and Scharaschkin.

I'm looking for families of curves for which the Belyi degree is ``easy to read off''. That is, a collection (finite or infinite) of smooth projective connected curves $X_i$ over $\bar{\mathbf{Q}}$ for which the Belyi degree can be bounded easily.

Are there any other nice examples?

J. Fac. Sci. Univ. Tokyo24(1977), 201-212. $\endgroup$ – Noam D. Elkies Jun 20 '11 at 16:52QKhadjavi and Scharaschkin show (Thm. 1a of myweb.lmu.edu/lkhadjavi/belyielliptic.pdf) that the curve $y^2 = x^3 + Ax + B$ has Belyi degree $O(|A|^3 + |B|^2)$. Similarly for a curve with full level-2 torsion and invariant $\lambda$, i.e. $cy^2 = x (x-1) (x-\lambda)$ for some rational $c,\lambda$ with $c \neq 0$ and $\lambda \neq 0, 1$, Belyi's construction gives an explicit Belyi map of degree $O(H(\lambda))$. Here $H$ is the height, $H(m/n) = \max(|m|,|n|)$. This is sharp when $H(\lambda)$ is prime, by a theorem of Beckmann (J.Alg. 125(1989), 236-255). $\endgroup$ – Noam D. Elkies Jun 20 '11 at 23:58