A noun and a participle in the genitive case and not modifying anything else in the sentence is a “genitive absolute”; see BBG 30.9 or this video.
You will translate genitive absolutes with a dependent clause.
The genitive noun will become the subject;
the genitive participle will be the verb.
The DMW will depend on the tense;
if present, use “while” or “because”;
if aorist, use “after”.
Ninety percent of genitive absolutes are temporal (GGBB 655).