sqrt5.sagews

from psage.number_fields.sqrt5.misc import *
from psage.number_fields.sqrt5.prime import *

p = primes_of_bounded_norm(10)[1]
p

5a

sp = p.sage_ideal()
sp.factor()

Fractional ideal (-2*a + 1)

sp._NumberFieldFractionalIdeal__factorization
#sage.structure.factorization.Factorization??
sp.factor() == sage.structure.factorization.Factorization([(sp,1)])

Fractional ideal (-2*a + 1) True

sp._pari_??

   File: /projects/sage/sage-7.5/local/lib/python2.7/site-packages/sage/rings/number_field/number_field_ideal.py
   Source:
       def _pari_(self):
        """
        Returns PARI Hermite Normal Form representations of this
        ideal.

        EXAMPLES::

            sage: K.<w> = NumberField(x^2 + 23)
            sage: I = K.class_group().0.ideal(); I
            Fractional ideal (2, 1/2*w - 1/2)
            sage: I._pari_()
            [2, 0; 0, 1]
        """
        return self.pari_hnf()