These curves have a direct interpretation as moduli spaces for elliptic curves with "level structure" and for this reason they play an important role in arithmetic geometry. The level "N" modular curve "X"("N") is the moduli space for elliptic curves with a basis for the "N"-torsion. For "X"0("N") and "X"1("N"), the level structure is, respectively, a cyclic subgroup of order "N" and a point of order "N". These curves have been studied in great detail, and in particular, it is known that "X"0("N") can be defined over Q.