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In this note we present a fast algorithm that finds for any $r$ the number$N_r$ of $\mathbb{F}_{q^r}$ rational points on a smooth absolutely irreduciblecurve $C$ defined over $\mathbb{F}_{q}$ assuming that we know $N_1,\cdots,N_g$,where $g$ is the genus of $C$. The proof of its validity is given in detail andits working are illustrated with several examples. In an Appendix we list thePython function in which we have implemented the algorithm together with otherroutines used in the examples.

A bootstrap for the number of $\mathbb{F}_{q^r}$-rational points on a curve over $\mathbb{F}_q$