src/sage/schemes/elliptic_curves: add "needs sage.libs.eclib"

Unsurprisingly, a lot of the tests in sage.schemes.elliptic_curves use
eclib. After a day of running sage -t followed by emacs followed by
sage -t followed by emacs... I think I've found them all. This commit
marks them as "needs sage.libs.eclib" so that they will be skipped
if eclib is unavailable or explicitly disabled.

Author: orlitzky

src/sage/schemes/elliptic_curves/BSD.py

@@ -86,6 +86,7 @@ def mwrank_two_descent_work(E, two_tor_rk) -> tuple:
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: from sage.schemes.elliptic_curves.BSD import mwrank_two_descent_work
         sage: E = EllipticCurve('14a')
         sage: mwrank_two_descent_work(E, E.two_torsion_rank())
@@ -130,7 +131,7 @@ def pari_two_descent_work(E) -> tuple:
         sage: pari_two_descent_work(E)
         (0, 0, 0, 0, [])
         sage: E = EllipticCurve('37a')
-        sage: pari_two_descent_work(E) # random, up to sign
+        sage: pari_two_descent_work(E) # random, up to sign, needs sage.libs.eclib
         (1, 1, 0, 0, [(0 : -1 : 1)])
         sage: E = EllipticCurve('210e7')
         sage: pari_two_descent_work(E)
@@ -215,6 +216,7 @@ def heegner_index_work(E) -> tuple:
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: from sage.schemes.elliptic_curves.BSD import heegner_index_work
         sage: heegner_index_work(EllipticCurve('14a'))
         (1, -31)
@@ -298,27 +300,31 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: EllipticCurve('11a').prove_BSD(verbosity=2)
         p = 2: True by 2-descent
         True for p not in {2, 5} by Kolyvagin.
         Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
         True for p = 5 by Kolyvagin bound
         []
 
+        sage: # needs sage.libs.eclib
         sage: EllipticCurve('14a').prove_BSD(verbosity=2)
         p = 2: True by 2-descent
         True for p not in {2, 3} by Kolyvagin.
         Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
         True for p = 3 by Kolyvagin bound
         []
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve("20a1")
         sage: E.prove_BSD(verbosity=2)
         p = 2: True by 2-descent
         True for p not in {2, 3} by Kolyvagin.
         Kato further implies that #Sha[3] is trivial.
         []
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve("50b1")
         sage: E.prove_BSD(verbosity=2)
         p = 2: True by 2-descent
@@ -337,16 +343,19 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     We know nothing with proof=True::
 
+        sage: # needs sage.libs.eclib
         sage: E.prove_BSD()
         Set of all prime numbers: 2, 3, 5, 7, ...
 
     We (think we) know everything with proof=False::
 
+        sage: # needs sage.libs.eclib
         sage: E.prove_BSD(proof=False)
         []
 
     A curve of rank 0 and prime conductor::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('19a')
         sage: E.prove_BSD(verbosity=2)
         p = 2: True by 2-descent
@@ -355,6 +364,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
         True for p = 3 by Kolyvagin bound
         []
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('37a')
         sage: E.rank()
         1
@@ -370,6 +380,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     We test the consistency check for the 2-part of Sha::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('37a')
         sage: S = E.sha(); S
         Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x
@@ -384,6 +395,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     An example with a Tamagawa number at 5::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('123a1')
         sage: E.prove_BSD(verbosity=2)
         p = 2: True by 2-descent
@@ -394,8 +406,9 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     A curve for which 3 divides the order of the Tate-Shafarevich group::
 
+        sage: # long time, needs sage.libs.eclib
         sage: E = EllipticCurve('681b')
-        sage: E.prove_BSD(verbosity=2)               # long time
+        sage: E.prove_BSD(verbosity=2)
         p = 2: True by 2-descent...
         True for p not in {2, 3} by Kolyvagin....
         Remaining primes:
@@ -406,6 +419,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     A curve for which we need to use ``heegner_index_bound``::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('198b')
         sage: E.prove_BSD(verbosity=1, secs_hi=1)
         p = 2: True by 2-descent
@@ -415,6 +429,7 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
     The ``return_BSD`` option gives an object with detailed information
     about the proof::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('26b')
         sage: B = E.prove_BSD(return_BSD=True)
         sage: B.two_tor_rk
@@ -432,21 +447,24 @@ def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
 
     This was fixed by :issue:`8184` and :issue:`7575`::
 
+        sage: # needs sage.libs.eclib
         sage: EllipticCurve('438e1').prove_BSD(verbosity=1)
         p = 2: True by 2-descent...
         True for p not in {2} by Kolyvagin.
         []
 
     ::
 
+        sage: # long time, needs sage.libs.eclib
         sage: E = EllipticCurve('960d1')
-        sage: E.prove_BSD(verbosity=1)  # long time (4s on sage.math, 2011)
+        sage: E.prove_BSD(verbosity=1)
         p = 2: True by 2-descent
         True for p not in {2} by Kolyvagin.
         []
 
     ::
 
+        sage: # needs sage.libs.eclib
         sage: E = EllipticCurve('66b3')
         sage: E.prove_BSD(two_desc="pari",verbosity=1)
         p = 2: True by 2-descent

src/sage/schemes/elliptic_curves/constructor.py

@@ -472,9 +472,11 @@ def create_object(self, version, key, *, names=None, **kwds):
 
         ``names`` is ignored at the moment, however it is used to support a convenient way to get a generator::
 
+            sage: # needs sage.libs.eclib
             sage: E.<P> = EllipticCurve(QQ, [1, 3])
             sage: P
             (-1 : 1 : 1)
+
             sage: E.<P> = EllipticCurve(GF(5), [1, 3])
             sage: P
             (4 : 1 : 1)
@@ -958,13 +960,15 @@ def EllipticCurve_from_cubic(F, P=None, morphism=True):
     the generator::
 
         sage: E = f.codomain()
+        sage: finv = f.inverse()
+
+        sage: # needs sage.libs.eclib
         sage: E.label()
         '720e2'
         sage: E.rank()
         1
         sage: R = E.gens()[0]; R
         (-17280000/2527 : 9331200000/6859 : 1)
-        sage: finv = f.inverse()
         sage: [finv(k*R) for k in range(1,10)]
         [(-4 : 1 : 0),
         (-1 : 4 : 1),
@@ -1469,6 +1473,7 @@ def EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: EllipticCurves_with_good_reduction_outside_S([])
         []
         sage: elist = EllipticCurves_with_good_reduction_outside_S([2])
@@ -1484,14 +1489,15 @@ def EllipticCurves_with_good_reduction_outside_S(S=[], proof=None, verbose=False
 
     Without ``Proof=False``, this example gives two warnings::
 
-        sage: elist = EllipticCurves_with_good_reduction_outside_S([11], proof=False)   # long time (14s on sage.math, 2011)
-        sage: len(elist)                                                                # long time
+        sage: # long time, needs sage.libs.eclib
+        sage: elist = EllipticCurves_with_good_reduction_outside_S([11], proof=False)
+        sage: len(elist)
         12
-        sage: ', '.join(e.label() for e in elist)                                       # long time
+        sage: ', '.join(e.label() for e in elist)
         '11a1, 11a2, 11a3, 121a1, 121a2, 121b1, 121b2, 121c1, 121c2, 121d1, 121d2, 121d3'
 
-        sage: # long time
-        sage: elist = EllipticCurves_with_good_reduction_outside_S([2,3])               # long time (26s on sage.math, 2011)
+        sage: # long time, needs sage.libs.eclib
+        sage: elist = EllipticCurves_with_good_reduction_outside_S([2,3])
         sage: len(elist)
         752
         sage: conds = sorted(set([e.conductor() for e in elist]))

src/sage/schemes/elliptic_curves/ell_egros.py

@@ -34,7 +34,8 @@
 EXAMPLES: We find all elliptic curves with good reduction outside 2,
 listing the label of each::
 
-    sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])]  # long time (5s on sage.math, 2013)
+    sage: # long time, needs sage.libs.eclib
+    sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([2])]
     ['32a1',
     '32a2',
     '32a3',
@@ -65,7 +66,8 @@
 installed: two of the auxiliary curves whose Mordell-Weil bases are
 required have conductors 13068 and 52272 so are in the database)::
 
-    sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)]  # long time (13s on sage.math, 2011)
+    sage: # long time, needs sage.libs.eclib
+    sage: [e.label() for e in EllipticCurves_with_good_reduction_outside_S([11], proof=False)]
     ['11a1', '11a2', '11a3', '121a1', '121a2', '121b1', '121b2', '121c1', '121c2', '121d1', '121d2', '121d3']
 
 AUTHORS:
@@ -319,6 +321,7 @@ def egros_from_jlist(jlist, S=[]):
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j, egros_from_jlist
         sage: jlist=egros_get_j([3])
         sage: elist=egros_from_jlist(jlist,[3])
@@ -371,14 +374,17 @@ def egros_get_j(S=[], proof=None, verbose=False):
 
     EXAMPLES::
 
+        sage: # needs sage.libs.eclib
         sage: from sage.schemes.elliptic_curves.ell_egros import egros_get_j
         sage: egros_get_j([])
         [1728]
-        sage: egros_get_j([2])  # long time (3s on sage.math, 2013)
+
+        sage: # long time, needs sage.libs.eclib
+        sage: egros_get_j([2])
         [128, 432, -864, 1728, 3375/2, -3456, 6912, 8000, 10976, -35937/4, 287496, -784446336, -189613868625/128]
-        sage: egros_get_j([3])  # long time (3s on sage.math, 2013)
+        sage: egros_get_j([3])
         [0, -576, 1536, 1728, -5184, -13824, 21952/9, -41472, 140608/3, -12288000]
-        sage: jlist=egros_get_j([2,3]); len(jlist) # long time (30s)
+        sage: jlist=egros_get_j([2,3]); len(jlist)
         83
     """
     if not all(p.is_prime() for p in S):

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -1511,7 +1511,7 @@ def gens(self):
             ...
             NotImplementedError: not implemented.
             sage: E = EllipticCurve(QQ, [1,1])
-            sage: E.gens()
+            sage: E.gens()  # needs sage.libs.eclib
             [(0 : 1 : 1)]
         """
         raise NotImplementedError("not implemented.")
@@ -2188,12 +2188,14 @@ def _multiple_x_numerator(self, n, x=None):
 
         ::
 
+            sage: # needs sage.libs.eclib
             sage: E = EllipticCurve("37a")
             sage: P = E.gens()[0]
             sage: x = P[0]
 
         ::
 
+            sage: # needs sage.libs.eclib
             sage: (35*P)[0]
             -804287518035141565236193151/1063198259901027900600665796
             sage: E._multiple_x_numerator(35, x)
@@ -2203,6 +2205,7 @@ def _multiple_x_numerator(self, n, x=None):
 
         ::
 
+            sage: # needs sage.libs.eclib
             sage: (36*P)[0]
             54202648602164057575419038802/15402543997324146892198790401
             sage: E._multiple_x_numerator(36, x)
@@ -2310,6 +2313,7 @@ def _multiple_x_denominator(self, n, x=None):
 
         ::
 
+            sage: # needs sage.libs.eclib
             sage: E = EllipticCurve("43a")
             sage: P = E.gens()[0]
             sage: x = P[0]