How can I have an internet social live in these conditions?

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It sounds as if your mother is a very responsible parent. Good for her!

If I were you, I would follow all rules that she has set for the phone. Do not abuse that privilege at all! In a couple of weeks, you might ask her if you can talk to her and explain that you have enjoyed the privilege of the phone, and that since you have followed all of the rules and used it wisely, you are wondering if you could have access to... and choose one of the above. I would likely choose Facebook first, as that is the most used. Tell her that you understand her concern and make sure she knows that 1. She can be on your friends list so that she can monitor you that way and 2. She can have the account password so she can monitor you that way. Also ask what rules she would have and, again, make sure to follow them. If she is not ready for Facebook yet, try Pinterest- it is the least dangerous for teenagers and she may be more open to that.