I am a 40cM DNA match (on MyHeritage) to one of Sophie's descendants Steven Lee Edwards (on MyHeritage)
My 3x great grandfather was Arnold Friedrich Ernst Arnold Friedrich Ernst
I'm not sure but based on the quality of the DNA match combined with the years they were born I would estimate that Arnold Friedrich Ernst would be an uncle of Sophie Ernst. I don't know anything about who Sophie's father may have been however.
Arnold's birthplace Brinkum (kingdom of Hanover) was probably not where his family originated because I have reason to think that his father (likely Sophie's grandfather) was in the Hanoverian Army at that time. Arnold's wedding record (Amsterdam 1847) says that his father's name was Joseph Ernst and his mother was "Maria Anna von Praag " (which is an odd combination of Dutch and German). This is just conjecture but I think Arnold's mother (possibly Sopie's grandmother) was a dutch Jew named "Marianne van Praag" who met non-Jewish Joseph Ernst while he was stationed in the Netherlands after the battle of Waterloo.