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Let us do the search explicitly. The multiplicative_order function cited in the question, applied on some object x delegates the work to the method with the same name of the object x, if any. In our case, this is - using a sample ring -

N = 202200002022
Z = IntegerModRing(N)
x = Z(101)
print(x.multiplicative_order())

We get

5882352

and the question is why. So let us ask for

sage: ??x.multiplicative_order

with the object x being the unit above, and we get the source and the location, here truncated info:

Source:   
    def multiplicative_order(self):
        """...
        """
        try:
            return sage.rings.integer.Integer(self.__pari__().znorder())
        except PariError:
            raise ArithmeticError("multiplicative order of %s not defined since it is not a unit modulo %s"%(
                self, self.__modulus.sageInteger))
File:      /usr/lib/python3.10/site-packages/sage/rings/finite_rings/integer_mod.pyx

So the code delegates the work to pari, and effectuates basicly

sage: x.__pari__()
Mod(101, 202200002022)
sage: x.__pari__().znorder()
5882352
sage: type(_)
<class 'cypari2.gen.Gen'>

and the obtained number is converted to a specific class, so that sage can handle it in the sequel.

Let us do the search explicitly. The multiplicative_order function cited in the question, applied on some object x delegates the work to the method with the same name of the object x, if any. In our case, this is - using a sample ring -

N = 202200002022
Z = IntegerModRing(N)
x = Z(101)
print(x.multiplicative_order())

We get

5882352

and the question is why. So let us ask for

sage: ??x.multiplicative_order

with the object x being the unit above, and we get the source and the location, here truncated info:

Source:   
    def multiplicative_order(self):
        """...
        """
        try:
            return sage.rings.integer.Integer(self.__pari__().znorder())
        except PariError:
            raise ArithmeticError("multiplicative order of %s not defined since it is not a unit modulo %s"%(
                self, self.__modulus.sageInteger))
File:      /usr/lib/python3.10/site-packages/sage/rings/finite_rings/integer_mod.pyx

So the code delegates the work to pari, and effectuates basicly

sage: x.__pari__()
Mod(101, 202200002022)
sage: x.__pari__().znorder()
5882352
sage: type(_)
<class 'cypari2.gen.Gen'>

and the obtained number is converted to a specific class, so that sage can handle it in the sequel.

[dan@k9 ~]$ gp
                            GP/PARI CALCULATOR Version 2.13.4 (released)
                    amd64 running linux (x86-64/GMP-6.2.1 kernel) 64-bit version
                          compiled: Apr  5 2022, gcc version 11.2.0 (GCC)
                                     threading engine: pthread
                           (readline v8.1 enabled, extended help enabled)

                               Copyright (C) 2000-2020 The PARI Group

PARI/GP is free software, covered by the GNU General Public License, and comes WITHOUT ANY WARRANTY 
WHATSOEVER.

Type ? for help, \q to quit.
Type ?17 for how to get moral (and possibly technical) support.

parisize = 8000000, primelimit = 500000, nbthreads = 8
? x = Mod(101, 202200002022)
%1 = Mod(101, 202200002022)
? znorder(x)
%2 = 5882352
?