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We review the Shimura-Taniyama method for computing the Newton polygon of anabelian variety with complex multiplication. We apply this method to cycliccovers of the projective line branched at three points. As an application, weproduce multiple new examples of Newton polygons that occur for Jacobians ofsmooth curves in characteristic $p$. Under certain congruence conditions on$p$, these include: the supersingular Newton polygon for each genus $g$ with $4\leq g \leq 11$; nine non-supersingular Newton polygons with $p$-rank $0$ with$4 \leq g \leq 11$; and, for all $g \geq 5$, the Newton polygon with $p$-rank$g-5$ having slopes $1/5$ and $4/5$.

Newton polygons of cyclic covers of the projective line branched at three points