  We explicitly bound the Faltings height of a curve over Q polynomially in its
Belyi degree. Similar bounds are proven for three other Arakelov invariants:
the discriminant, Faltings' delta invariant and the self-intersection of the
dualizing sheaf. Our results allow us to explicitly bound Arakelov invariants
of modular curves, Hurwitz curves and Fermat curves in terms of their genus.
Moreover, as an application, we show that the Couveignes-Edixhoven-Bruin
algorithm to compute coefficients of modular forms for congruence subgroups of
SL2(Z) runs in polynomial time under the Riemann hypothesis for zeta-functions
of number fields. This was known before only for certain congruence subgroups.
Finally, we use our results to prove a conjecture of Edixhoven, de Jong and
Schepers on the Faltings height of a cover of the projective line with fixed
branch locus. Our proof uses Merkl's method for bounding Arakelov-Green's
functions to deal with archimedean contributions. In the appendix, Peter Bruin
proves an explicit version of Merkl's results.
