Does the shaving head fit on the Braun 5770? Thank you very much!

I can't answer that, I have other Braun razors from the 5 series.

Hello!
When I bought the shaver, there was a table on the BRAUN website. Unfortunately, I left my shaver behind on holiday. But a quick search showed that the 31B foil should almost certainly fit the 5414. I don't want to post competitor links here. Search for "Braun 5414 shear blade". There are a lot of pages where you can see that this is the right blade.
Have fun with it and smooth skin all round ;-)