Populations of rotating stars III. SYCLIST, the new Geneva population synthesis code

Context. Constraints on stellar models can be obtained from observations of stellar populations, provided the population results from a well defined star formation history. Aims. We present a new tool for building synthetic colour-magnitude diagrams of coeval stellar populations. We study, from a theoretical point of view, the impact of axial rotation of stars on various observed properties of single-aged stellar populations: magnitude at the turnoff, photometric properties of evolved stars, surface velocities, surface abundances, and the impact of rotation on the age determination of clusters by an isochrone fitting. One application to the cluster NGC 663 is performed. Methods. Stellar models for different initial masses, metallicities, and zero-age main sequence (ZAMS) rotational velocities are used for building interpolated stellar tracks, isochrones, and synthetic clusters for various ages and metallicities. The synthetic populations account for the effects of the initial distribution of the rotational velocities on the ZAMS, the impact of the inclination angle and the effects of gravity and limb darkening, unresolved binaries, and photometric errors. Interpolated tracks, isochrones, and synthetic clusters can be computed through a public web interface. Results. For clusters with a metallicity in the range [0 : 002; 0 : 014] and an age between 30 Myr and 1 Gyr, the fraction of fast rotators on the main sequence (MS) band is the largest just below the turnoff. This remains true for two different published distributions of the rotational velocities on the ZAMS. This is a natural consequence of the increase in the MS lifetime due to rotation. The fraction of fast rotators one magnitude below the turnoff also increases with the age of the cluster between 30 Myr and 1 Gyr. The most nitrogen-rich stars are found just below the turnoff. There is an increase in the fraction of enriched stars when the metallicity decreases. We show that the use of isochrones computed from rotating stellar models with an initial rotation that is representative of the average initial rotation of the stars in clusters provides a reasonable estimate of the age, even though stars in a real cluster did not start their evolution with an identical initial rotation.