Have got to (/hæv gɒʔ tu:/) is an English semi-modal verb. It uses the have and the perfect aspect with the past participle of get, followed by a to-infinitive to express obligation.
The meaning is obligation. The negative "haven't got to" means you can do it if you want, so is different to musnt't.
This conjugates with the 3rd person -s in the present tense. The past tense form is "had got to".
For this usage the past participle used is always "got" rather than "gotten".
The have in "have got to" is subject to vowel reduction to /həv/ - with h-dropping /əv/ - and may be ellided completely. That leaves non-standard "gotta".
This form is common in registers of spoken English, and informal written English but is less appropriate in, and often proscribed from, formal written English where have to or must are generally preferred.