We provide the complete four loop perturbative renormalization of a low temperature statistical mechanics model of flat polymerized membranes. Within a non-local effective flexural theory based on transverse elastic fluctuations, we determine analytically, at the four existing fixed points, the anomalous elasticity critical exponent $\eta$ which controls all scaling behaviors in the theory. Results are obtained as apparently convergent series, enabling precise estimates without resummations. We independently confirm and complement the results of recent four-loop work [Pikelner 2022 EPL 138 17002] carried out in a different model. We also find good agreement with non-perturbative approaches and experiments on soft materials and graphene.