A computation shows that there are
77
77
(up to scalar shifts) possible pairs of integer coefficient polynomials of degree five having roots of unity as their roots and satisfying the conditions of Beukers and Heckman, so that the Zariski closures of the associated monodromy groups are either finite or the orthogonal groups of non-degenerate and non-positive quadratic forms. Following the criterion of Beukers and Heckman, it is easy to check that only
4
4
of these pairs correspond to finite monodromy groups, and only
17
17
pairs correspond to monodromy groups, for which the Zariski closure has real rank one.
There are
56
56
pairs remaining, for which the Zariski closures of the associated monodromy groups have real rank two. It follows from Venkataramana that
11
11
of these
56
56
pairs correspond to arithmetic monodromy groups, and the arithmeticity of
2
2
other cases follows from Singh. In this article, we show that
23
23
of the remaining
43
43
rank two cases correspond to arithmetic groups.