gh-35787: adjust random tests for generators of elliptic curves

#35626 introduced a test failure on some machines as some tests return
random choices of generators of Mordell-Weil groups for elliptic curves.

This adds "random" to these tests and add some tests that do test if the
returned answer is correct. This is only possible when the rank is 1
(and the returned point must be in a finite set). For larger ranks at
best we can test if the number of generators is ok and if the points are
saturated.

URL: #35787
Reported by: Chris Wuthrich
Reviewer(s):

Author: Release Manager

build/pkgs/configure/checksums.ini

@@ -1,4 +1,4 @@
 tarball=configure-VERSION.tar.gz
-sha1=1def339a9f1b9f34ee5fb82f38a95914876618d8
-md5=520b4c716ee0ebc94fbb5d81f17efa85
-cksum=2018559679
+sha1=310e1602a96449fc1da190278048e89f63c71ef2
+md5=20faa4d933ea72b0fc3ab323c8d5cc97
+cksum=2486899926

build/pkgs/configure/package-version.txt

@@ -1 +1 @@
-c418a82d8985d727cb63c5d3f62c60264b52fd13
+a198b0e946fab4b5ff3b0a2bdc1dbad86f24d24c

src/sage/schemes/elliptic_curves/ell_rational_field.py

@@ -2319,6 +2319,22 @@ def gens(self, proof=None, **kwds):
             sage: E1.gens(algorithm="pari")   #random
             [(-400 : 8000 : 1), (0 : -8000 : 1)]
 
+        TESTS::
+
+            sage: E = EllipticCurve('389a')
+            sage: len(E.gens())
+            2
+            sage: E.saturation(E.gens())[1]
+            1
+            sage: len(E.gens(algorithm="pari"))
+            2
+            sage: E.saturation(E.gens(algorithm="pari"))[1]
+            1
+            sage: E = EllipticCurve([-3/8,-2/3])
+            sage: P = E.lift_x(10/9)
+            sage: set(E.gens()) <= set([P,-P])
+            True
+
         """
         if proof is None:
             from sage.structure.proof.proof import get_flag
@@ -2370,23 +2386,29 @@ def _compute_gens(self, proof,
             sage: proved
             True
 
-        TESTS::
-
             sage: E = EllipticCurve([-127^2,0])
-            sage: E.gens(use_database=False, algorithm="pari")
-            Traceback (most recent call last):
-            ...
-            RuntimeError: generators could not be determined. So far we found []. Hint: increase pari_effort.
-            sage: E.gens(use_database=False, algorithm="pari",pari_effort=4)
+            sage: E.gens(use_database=False, algorithm="pari",pari_effort=4)   # random
             [(611429153205013185025/9492121848205441 : 15118836457596902442737698070880/924793900700594415341761 : 1)]
 
+        TESTS::
+
+            sage: P = E.lift_x(611429153205013185025/9492121848205441)
+            sage: set(E.gens(use_database=False, algorithm="pari",pari_effort=4)) <= set([P+T for T
+            ....:  in E.torsion_points()] + [-P+T for T in E.torsion_points()])
+            True
+
             sage: E = EllipticCurve([-157^2,0])
             sage: E.gens(use_database=False, algorithm="pari")
             Traceback (most recent call last):
             ...
             RuntimeError: generators could not be determined. So far we found []. Hint: increase pari_effort.
-            sage: E.gens(use_database=False, algorithm="pari",pari_effort=10) # long time
+            sage: ge = E.gens(use_database=False, algorithm="pari",pari_effort=10)
+            sage: ge   #random
             [(-166136231668185267540804/2825630694251145858025 : 167661624456834335404812111469782006/150201095200135518108761470235125 : 1)]
+            sage: P = E.lift_x(-166136231668185267540804/2825630694251145858025)
+            sage: set(E.gens(use_database=False, algorithm="pari",pari_effort=4)) <= set([P+T for T
+            ....:  in E.torsion_points()] + [-P+T for T in E.torsion_points()])
+            True
 
         """
         # If the optional extended database is installed and an
@@ -2532,6 +2554,11 @@ def gens_certain(self):
 
         TESTS::
 
+            sage: E = EllipticCurve('37a1')
+            sage: P = E([0,-1])
+            sage: set(E.gens()) <= set([P,-P])
+            True
+
             sage: E = EllipticCurve([2, 4, 6, 8, 10])
             sage: E.gens_certain()
             Traceback (most recent call last):