The following question was motivated by one of the earliest exercises of [Complex Abelian Variaties by Birkenhake and Lange](http://books.google.ca/books/about/Complex_Abelian_varieties.html?id=MOW2gEP7HIkC) during my presentation last year. 

It can be shown that any complex torus $X$ $(=V/\Lambda$, where $V$ is a complex vector space and $\Lambda$ is lattice of maximal real dimension in $V)$ admits at most countably many complex subtori.

My question:

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> Is there sort of algorithm s.t. one could find simple (not admitting any non-trivial complex subtorus) complex tori of dimension $\geq 2?$ how about simple abelian varieties of dimension $\geq 2?$ 
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Note that, $X$ admits a complex subtorus of dimension $g'$ if and only if there exists a subgroup $\Lambda' \subset \Lambda$ of rank $2g'$ s.t. the image of the canonical map $\Lambda' \otimes \mathbb{R} \to V$ is a complex subvector space of $V.$