Improve construction of narrow class groups

src/sage/rings/number_field/class_group.py

@@ -321,8 +321,20 @@
         """
         return self.ideal().gens()
 
+class NarrowFractionalIdealClass(FractionalIdealClass):
+    def _repr_(self):
+        r"""
+        Return a string representation of the narrow ideal class of this fractional ideal.
+
+        EXAMPLES::
+
+        """
+        if self.is_trivial():
+            return 'Trivial narrow ideal class'
+        return 'Fractional narrow ideal class %s' % self._value._repr_short()
+    pass
 
 class SFractionalIdealClass(FractionalIdealClass):
     r"""
     An `S`-fractional ideal class in a number field for a tuple `S` of primes.
 
@@ -628,8 +640,26 @@
         """
         return self._number_field
 
+class NarrowClassGroup(ClassGroup):
+    Element = NarrowFractionalIdealClass
+
+    def __init__(self, gens_orders, names, number_field, gens, proof=True):
+        r"""
+        Create a narrow class group.
+        """
+        AbelianGroupWithValues_class.__init__(self, gens_orders, names, gens,
+                                              values_group=number_field.ideal_monoid())
+        self._proof_flag = proof
+        self._number_field = number_field
+
+    def _repr_(self):
+        s = 'Narrow class group of order %s ' % self.order()
+        if self.order() > 1:
+            s += 'with structure %s ' % self._group_notation(self.gens_orders())
+        s += 'of %s' % self.number_field()
+        return s
 
 class SClassGroup(ClassGroup):
     r"""
     The `S`-class group of a number field.
 

src/sage/rings/number_field/number_field.py

@@ -6706,7 +6706,7 @@
         return S
 
     @cached_method
-    def narrow_class_group(self, proof=None):
+    def narrow_class_group(self, proof=None, names='c'):
         r"""
         Return the narrow class group of this field.
 
@@ -6719,19 +6719,32 @@
 
             sage: x = polygen(QQ, 'x')
             sage: NumberField(x^3 + x + 9, 'a').narrow_class_group()
-            Multiplicative Abelian group isomorphic to C2
+            Narrow class group of order 2 with structure C2 of Number Field in a with defining polynomial x^3 + x + 9
 
         TESTS::
 
             sage: QuadraticField(3, 'a').narrow_class_group()
-            Multiplicative Abelian group isomorphic to C2
+            Narrow class group of order 2 with structure C2 of Number Field in a with defining polynomial x^2 - 3 with a = 1.732050807568878?
         """
-        from sage.groups.abelian_gps.abelian_group import AbelianGroup
+        from .class_group import NarrowClassGroup
 
         proof = proof_flag(proof)
+        try:
+            return self.__narrow_class_group[proof, names]
+        except KeyError:
+            pass
+        except AttributeError:
+            self.__narrow_class_group = {}
         k = self.pari_bnf(proof)
-        s = k.bnfnarrow().sage()
-        return AbelianGroup(s[1])
+        s = k.bnfnarrow()
+        cycle_structure = tuple(s[1].sage())
+
+        # Gens is a list of ideals (the generators)
+        gens = tuple(self.ideal(hnf) for hnf in s[2])
+
+        G = NarrowClassGroup(cycle_structure, names, self, gens, proof=proof)
+        self.__narrow_class_group[proof, names] = G
+        return G
 
     def ngens(self):
         """