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I don't think this question is going to have a GREAT answer -- your examples 1 and 2 are HANDED to you as Belyi covers of the line, and I'd think any family that doesn't immediately present itself in this way is unlikely to offer an easy upper bound on Belyi degree. But that's not an answer, so here's one more -- any Hurwitz curve parametrizing covers of P^1 branched at four points will have a Belyi map (namely, the map to M_{0,4}) whose degree you can read off quite directly. |
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I don't think this question is going to have a GREAT answer -- your examples 1 and 2 are HANDED to you as Belyi covers of the line, and I'd think any family that doesn't immediately present itself in this way is unlikely to offer an easy upper bound on Belyi degree. But that's not an answer, so here's one more -- any Hurwitz curve parametrizing covers of P^1 branched at four points will have Belyi map (namely, the map to M_{0,4}) whose degree you can read off quite directly. |
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