change sorting order for WeierstrassIsomorphisms

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -2430,6 +2430,9 @@ def automorphisms(self, field=None):
         """
         Return the set of isomorphisms from self to itself (as a list).
 
+        The identity and negation morphisms are guaranteed to appear
+        as the first and second entry of the returned list.
+
         INPUT:
 
         - ``field`` (default ``None``) -- a field into which the
@@ -2447,23 +2450,52 @@ def automorphisms(self, field=None):
             sage: E = EllipticCurve_from_j(QQ(0))  # a curve with j=0 over QQ
             sage: E.automorphisms()
             [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field
-            Via:  (u,r,s,t) = (-1, 0, 0, -1), Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field
-            Via:  (u,r,s,t) = (1, 0, 0, 0)]
+               Via:  (u,r,s,t) = (1, 0, 0, 0),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field
+               Via:  (u,r,s,t) = (-1, 0, 0, -1)]
 
         We can also find automorphisms defined over extension fields::
 
             sage: K.<a> = NumberField(x^2+3)  # adjoin roots of unity
             sage: E.automorphisms(K)
             [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
-            Via:  (u,r,s,t) = (-1, 0, 0, -1),
-            ...
-            Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
-            Via:  (u,r,s,t) = (1, 0, 0, 0)]
+               Via:  (u,r,s,t) = (1, 0, 0, 0),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
+               Via:  (u,r,s,t) = (-1, 0, 0, -1),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
+               Via:  (u,r,s,t) = (-1/2*a - 1/2, 0, 0, 0),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
+               Via:  (u,r,s,t) = (1/2*a + 1/2, 0, 0, -1),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
+               Via:  (u,r,s,t) = (1/2*a - 1/2, 0, 0, 0),
+             Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Number Field in a with defining polynomial x^2 + 3
+               Via:  (u,r,s,t) = (-1/2*a + 1/2, 0, 0, -1)]
 
         ::
 
             sage: [len(EllipticCurve_from_j(GF(q,'a')(0)).automorphisms()) for q in [2,4,3,9,5,25,7,49]]
             [2, 24, 2, 12, 2, 6, 6, 6]
+
+        TESTS:
+
+        Random testing::
+
+            sage: p = random_prime(100)
+            sage: k = randrange(1,30)
+            sage: F.<t> = GF((p,k))
+            sage: while True:
+            ....:     try:
+            ....:         E = EllipticCurve(list((F^5).random_element()))
+            ....:     except ArithmeticError:
+            ....:         continue
+            ....:     break
+            sage: Aut = E.automorphisms()
+            sage: Aut[0] == E.multiplication_by_m_isogeny(1)
+            True
+            sage: Aut[1] == E.multiplication_by_m_isogeny(-1)
+            True
+            sage: sorted(Aut) == Aut
+            True
         """
         if field is not None:
             self = self.change_ring(field)
@@ -2493,9 +2525,9 @@ def isomorphisms(self, other, field=None):
             sage: F = EllipticCurve('27a3') # should be the same one
             sage: E.isomorphisms(F)
             [Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field
-              Via:  (u,r,s,t) = (-1, 0, 0, -1),
+               Via:  (u,r,s,t) = (1, 0, 0, 0),
              Elliptic-curve endomorphism of Elliptic Curve defined by y^2 + y = x^3 over Rational Field
-              Via:  (u,r,s,t) = (1, 0, 0, 0)]
+               Via:  (u,r,s,t) = (-1, 0, 0, -1)]
 
         We can also find isomorphisms defined over extension fields::
 

src/sage/schemes/elliptic_curves/weierstrass_morphism.py

@@ -32,6 +32,32 @@
 from sage.structure.sequence import Sequence
 from sage.rings.all import Integer, PolynomialRing
 
+def _urst_sorting_key(tup):
+    r"""
+    Return a sorting key for `(u,r,s,t)` tuples representing
+    elliptic-curve isomorphisms. The key is chosen in such a
+    way that an isomorphism and its negative appear next to
+    one another in a sorted list, and such that normalized
+    isomorphisms come first. One particular consequence of
+    this is that the identity and negation morphisms are the
+    first and second entries of the list returned by
+    :meth:`~sage.schemes.elliptic_curves.ell_generic.EllipticCurve_generic.automorphisms`.
+
+    TESTS::
+
+        sage: from sage.schemes.elliptic_curves.weierstrass_morphism import _urst_sorting_key
+        sage: _urst_sorting_key((1,0,0,0)) < _urst_sorting_key((-1,0,0,0))
+        True
+        sage: _urst_sorting_key((-1,0,0,0)) < _urst_sorting_key((2,0,0,0))
+        True
+        sage: _urst_sorting_key((1,2,3,4)) < _urst_sorting_key((-1,0,0,0))
+        True
+    """
+    v = tup[0]
+    h = 0 if v == 1 else 1 if v == -1 else 2
+    if -v < v:
+        v = -v
+    return (h, v) + tup
 
 @richcmp_method
 class baseWI():
@@ -91,12 +117,7 @@ def __richcmp__(self, other, op):
         """
         Standard comparison function.
 
-        The ordering is just lexicographic on the tuple `(u,r,s,t)`.
-
-        .. NOTE::
-
-            In a list of automorphisms, there is no guarantee that the
-            identity will be first!
+        The ordering is done according to :func:`_urst_sorting_key`.
 
         EXAMPLES::
 
@@ -117,7 +138,9 @@ def __richcmp__(self, other, op):
         """
         if not isinstance(other, baseWI):
             return op == op_NE
-        return richcmp(self.tuple(), other.tuple(), op)
+        key1 = _urst_sorting_key(self.tuple())
+        key2 = _urst_sorting_key(other.tuple())
+        return richcmp(key1, key2, op)
 
     def tuple(self):
         r"""
@@ -543,7 +566,7 @@ def _comparison_impl(left, right, op):
             sage: w1 = E.isomorphism_to(F)
             sage: w1 == w1
             True
-            sage: w2 = F.automorphisms()[0] * w1
+            sage: w2 = F.automorphisms()[1] * w1
             sage: w1 == w2
             False