Trac #16011: Fix _torsion_bound() for elliptic curves over relative n…

…umber fields

Computing torsion subgroups of elliptic curves over relative number
fields fails:
{{{
sage: R.<x> = QQ[]
sage: F.<a> = QuadraticField(5)
sage: K.<b> = F.extension(x^2-3)
sage: E = EllipticCurve(K,[0,0,0,b,1])
sage: E.torsion_subgroup()
...
NotImplementedError: For an ideal in a relative number field you must
use relative_ramification_index or absolute_ramification_index as
appropriate
}}}
The solution should be to make `E._torsion_bound()` use
`absolute_ramification_index()` instead of `ramification_index()`.

URL: http://trac.sagemath.org/16011
Reported by: pbruin
Ticket author(s): Marc Masdeu
Reviewer(s): Peter Bruin

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -1657,6 +1657,16 @@ def _torsion_bound(self,number_of_places = 20):
             [6, 6, 6, 6, 6, 6]
             sage: [E.torsion_order() for E in CDB.iter([14])]
             [6, 6, 2, 6, 2, 6]
+
+        An example over a relative number field (see :trac:`16011`)::
+
+            sage: R.<x> = QQ[]
+            sage: F.<a> = QuadraticField(5)
+            sage: K.<b> = F.extension(x^2-3)
+            sage: E = EllipticCurve(K,[0,0,0,b,1])
+            sage: E.torsion_subgroup().order()
+            1
+
         """
         E = self
         bound = 0
@@ -1678,7 +1688,7 @@ def _torsion_bound(self,number_of_places = 20):
                 if fqq == 1 or charqq**fqq < 3*number_of_places:
                     # check if the model is integral at the place
                     if min([K.ideal(a).valuation(qq) for a in E.a_invariants()]) >= 0:
-                        eqq = qq.ramification_index()
+                        eqq = qq.absolute_ramification_index()
                         # check if the formal group at the place is torsion-free
                         # if so the torsion injects into the reduction
                         if eqq < charqq - 1 and disc.valuation(qq) == 0: