If you can check that the curve is geometrically irreducible, then you may try using the Hurwitz formula (you may use the formula in any case, but you would have to be more careful with the conclusions if the curve is not irreducible). Assuming that your example is geometrically integral, projection onto one of the axis is a morphism of degree 23. If you know that the morphism ramifies in at least 2*23+h smooth points, then the curve has genus at least h/2. If you know the multiplicities of the ramification or if you know the behaviour of ramification at the singular points, then you can deduce more accurate information.
damiano
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