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It fell to the floor from a low height, and I tried to press all the buttons, together, all kinds times.
It just does not turn on.

i think you should give it to sony for maintenance..cause such a phone is very sensitve..u should get some kinda flip cover and a screen protector 
i have a cover and a screen protector
And it has been dropped several times before and nothing happened
As per Empath, I would advise contacting your local Xperia Care team as this will likely need to be assessed by an authorised repair agent.
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