Ag.algebraic geometry – What integer value can be the driver of an abelian variety to $ g $ more than $ mathbb Q $?

Set a positive integer $ g $. What a positive integer $ n $ can be the driver of a $ g $three-dimensional abelian variety on $ mathbb Q $ ?

For example, as there are no abelian varieties on $ mathbb Z $, $ N $ it is not possible $ 1 $. And for elliptical curves, $ N $ must not be less than $ 11 $.