  In perturbative quantum field theory one encounters certain, very specific
geometries over the integers. These perturbative quantum geometries determine
the number contents of the amplitude considered. In the article `Modular forms
in quantum field theory' F. Brown and the author report on a first list of
perturbative quantum geometries using the $c_2$-invariant in $\phi^4$ theory. A
main tool was denominator reduction which allowed the authors to examine graphs
up to loop order (first Betti number) 10. We introduce an improved quadratic
denominator reduction which makes it possible to extend the previous results to
loop order 11 (and partially orders 12 and 13). For comparison, also
non-$\phi^4$ graphs are investigated. Here, we extend the results from loop
order 9 to 10. The new database of 4801 unique $c_2$-invariants (previously
157) -- while being consistent with all major $c_2$-conjectures -- leads to a
more refined picture of perturbative quantum geometries. In the appendix,
Friedrich Knop proves a Chevalley-Warning-Ax theorem for double covers of
affine space.
