Abstract
We use class field theory to search for curves with many rational points over the finite fields of cardinality ⩽5. By going through abelian covers of each curve of genus ⩽2 over these fields we find a number of new curves. In particular, over F2 we settle the question of how many points there can be on a curve of genus 17 by finding one with 18 points. The search is aided by computer; in some cases it is exhaustive for this type of curve of genus up to 50.
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