Over 30mins to generate the first eigenvector?

Avoiding exact calculation by using approximated numerical value before calculation speeds things up. I'm also calculating only the first Eigenvectors as pointed pot by @Öskå. Now it takes only 15 milliseconds time

AbsoluteTiming[Chop@Eigenvectors[N[m], 1, Quartics -> True]]

{0.015600, {{-0.0725514, -0.106358, -0.0986766, -0.110735 [...] }}}

By using a sparse solver you can gain another factor of ten in speed over @rhermans' method:

Eigenvectors[m // SparseArray // N, 1, Method -> "Arnoldi"] // AbsoluteTiming

(*    {0.001119, {{-0.0725514, -0.106358, -0.0986766, -0.110735, [...]}}}   *)