interger-ring() and maximal_order() is the same in numberfield

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I think you must mean ring_of_integers rather than "interger-ring". And indeed K.ring_of_integers() and K.maximal_order() are the same for a number field K. You can see this explicitly as follows:

sage: K.ring_of_integers??
Type:       builtin_function_or_method
String Form:<built-in method ring_of_integers of NumberField_absolute_with_category object at     0x7f85bc4dae00>
Definition: K.ring_of_integers(self, *args, **kwds)
Source:
    def ring_of_integers(self, *args, **kwds):
        r"""
        Synomym for ``self.maximal_order(...)``.

        EXAMPLES::

            sage: K.<a> = NumberField(x^2 + 1)
            sage: K.ring_of_integers()
            Maximal Order in Number Field in a with defining polynomial x^2 + 1
        """
       return self.maximal_order()