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We investigate low-degree points on the Fermat curve of degree 13, the Snyder quintic curve and the Klein quartic curve. We compute all quadratic points on these curves and use Coleman’s effective Chabauty method to obtain bounds for the number of cubic points on each of the former two curves.

We describe the group generated by the Weierstrass points in the Jacobian of the curve X 4 +Y 4 +Z 4 +3 (X 2 Y 2 +X 2 Z 2 +Y 2 Z 2 )) = 0 This curve is the only curve of genus 3, apart from the fourth Fermat curve, possessing exactly twelve Weierstrass points.

Let J 5 J_{5} denote the Jacobian of the Fermat curve of exponent 5 and let K = Q ( ζ 5 ) K=Q(\zeta _{5}) . We compute the groups J 5 ( K ) J_{5}(K) , J 5 ( K + ) J_{5}(K^{+}) , J 5 ( Q ) J_{5}(Q) , where K + K^{+} is the unique quadratic subfield of K K . As an application, we present a new proof that there are no K K -rational points on the 5-th Fermat curve, except the so called “points at infinity".