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A slightly more theoretical answer: there is an algorithm of Tate, called (unremarkably) Tate's algorithm, which allows one to compute the minimal model over any local field. I have a vague memory that it's not proved that this algorithm terminates in general, although it is expect expected to. (Perhaps someone else can say something definitive about this.) I would guess that this is what is implemented in Pari.

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A slightly more theoretical answer: there is an algorithm of Tate, called (unremarkably) Tate's algorithm, which allows one to compute the minimal model over any local field. I have a vague memory that it's not proved that this algorithm terminates in general, although it is expect to. (Perhaps someone else can say something definitive about this.) I would guess that this is what is implemented in Pari.