Have got to
This conjugates with the 3rd person -s in the present tense. The past tense form is "had got to". It's possible to create a perfect gerund from this, e.g. "Having got to do something,", but it's not possible to use a continuous aspect - *"I have been getting 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. The /u:/ meanwhile can also be reduced to a schwa. That leaves non-standard "gotta". The /t/ sound in "got to" is subject to sound twinning and may be reduced to a glottal stop. Americans often flap the /t/ so it sounds like "godda".
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.