Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group.
[clarification needed] Hodge diamond: Any hyperelliptic surface is a quotient (E×F)/G, where E = C/Λ and F are elliptic curves, and G is a subgroup of F (acting on F by translations), which acts on E not only by translations.
There are seven families of hyperelliptic surfaces as in the following table.
A quasi-hyperelliptic surface is a surface whose canonical divisor is numerically equivalent to zero, the Albanese mapping maps to an elliptic curve, and all its fibers are rational with a cusp.
Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).