Trac #16022: simon_two_descent sometimes gives wrong answers

The current version of Denis Simon's 2-descent program (in Sage since
#11005) has a bug:
{{{
sage: K.<y> = NumberField(x^4 + x^2 - 7);
sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0])
sage: E.simon_two_descent(lim1=2, limtriv=3)
[0, 0, []]
}}}
The rank is in fact 1, and a correct return value would be `(1, 1,
[(-369/25*y^3 + 539/25*y^2 - 1178/25*y + 1718/25 : -27193/125*y^3 +
39683/125*y^2 - 86816/125*y + 126696/125 : 1)])`.

This ticket is just to fix this as quickly as possible; it should
eventually be fixed in a new version of Simon's program.

(See #15608 for a list of tickets related to `simon_two_descent`.)

URL: http://trac.sagemath.org/16022
Reported by: pbruin
Ticket author(s): Peter Bruin
Reviewer(s): Marc Masdeu

src/ext/pari/simon/ell.gp

@@ -1776,8 +1776,11 @@ if( DEBUGLEVEL_ell >= 2, print("  reduced: Y^2 = ",lift(redq[1])));
 \\ Search for a point on the quartic
       point = nfratpoint(nf,pol,LIM1,1);
       found = point != [];
-\\ If no point is found, check if it is ELS
-      if( !found && !loc,
+      if( found,
+        loc = 1
+      );
+\\ If the quartic is not known to be ELS, check if it is
+      if( !loc,
         if( bigflag,
           loc = nflocallysoluble(nf,pol,r,a,b)
         , loc = nflocallysoluble(nf,pol,0,1,1)

src/sage/schemes/elliptic_curves/gp_simon.py

@@ -60,6 +60,13 @@ def simon_two_descent(E, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, lim
         sage: E.simon_two_descent()
         (1, 1, [(-1 : 0 : 1)])
 
+    Check that :trac:`16022` is fixed::
+
+        sage: K.<y> = NumberField(x^4 + x^2 - 7);
+        sage: E = EllipticCurve(K, [1, 0, 5*y^2 + 16, 0, 0])
+        sage: E.simon_two_descent(lim1=2, limtriv=3)  # long time (about 3 s)
+        (1, 1, [(-369/25*y^3 + 539/25*y^2 - 1178/25*y + 1718/25 : -27193/125*y^3 + 39683/125*y^2 - 86816/125*y + 126696/125 : 1)])
+
     """
     init()