Fano surfaces with 12 or 30 elliptic curves

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N'eron-Severi group of the Fano surface of the Fermat cubic threefold.