Trac #17295: For elliptic curves over relative number fields, is_isog…

…enous() raises an error

The method `EllipticCurve_number_field.is_isogenous()` computes norms of
ideals of the base field using the `norm()` method.  To make this work
in the case where the base field is a relative number field, one has to
use the `absolute_norm()` method instead:
{{{
sage: k.<s> = QuadraticField(2)
sage: K.<b> = k.extension(x^2 - 3)
sage: E = EllipticCurve(k, [-3*s*(4 + 5*s), 2*s*(2 + 14*s + 11*s^2)])
sage: Ec = EllipticCurve(k, [3*s*(4 - 5*s), -2*s*(2 - 14*s + 11*s^2)])
sage: EK = E.base_extend(K)
sage: EcK = Ec.base_extend(K)
sage: EK.is_isogenous(EcK)
Traceback (most recent call last):
...
NotImplementedError: For a fractional ideal in a relative number field
you must use relative_norm or absolute_norm as appropriate
}}}

URL: http://trac.sagemath.org/17295
Reported by: pbruin
Ticket author(s): Peter Bruin
Reviewer(s): Frédéric Chapoton

src/sage/schemes/elliptic_curves/ell_number_field.py

@@ -2727,6 +2727,17 @@ def is_isogenous(self, other, proof=True, maxnorm=100):
             sage: E.is_isogenous(Ec)
             True
 
+        Check that :trac:`17295` is fixed::
+
+            sage: k.<s> = QuadraticField(2)
+            sage: K.<b> = k.extension(x^2 - 3)
+            sage: E = EllipticCurve(k, [-3*s*(4 + 5*s), 2*s*(2 + 14*s + 11*s^2)])
+            sage: Ec = EllipticCurve(k, [3*s*(4 - 5*s), -2*s*(2 - 14*s + 11*s^2)])
+            sage: EK = E.base_extend(K)
+            sage: EcK = Ec.base_extend(K)
+            sage: EK.is_isogenous(EcK)      # long time (about 3.5 s)
+            True
+
         """
         if not is_EllipticCurve(other):
             raise ValueError("Second argument is not an Elliptic Curve.")
@@ -2745,7 +2756,7 @@ def is_isogenous(self, other, proof=True, maxnorm=100):
         PI = K.primes_of_degree_one_iter()
         while True:
             P = next(PI)
-            if P.norm() > maxnorm: break
+            if P.absolute_norm() > maxnorm: break
             if not P.divides(N):
                 if E1.reduction(P).cardinality() != E2.reduction(P).cardinality():
                     return False
@@ -2770,7 +2781,7 @@ def is_isogenous(self, other, proof=True, maxnorm=100):
         # Next we try looking modulo some more primes:
 
         while True:
-            if P.norm() > 10*maxnorm: break
+            if P.absolute_norm() > 10*maxnorm: break
             if not P.divides(N):
                 OP = K.residue_field(P)
                 if E1.change_ring(OP).cardinality() != E2.change_ring(OP).cardinality():