Have got to
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 reduction 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.