bash-2.05a$ mwrank3
Program mwrank: uses 2-descent (via 2-isogeny if possible) to
determine the rank of an elliptic curve E over Q, and list a
set of points which generate E(Q) modulo 2E(Q).
and finally search for further points on the curve.
For more details see the file mwrank.doc.
For details of algorithms see the author's book.

Please acknowledge use of this program in published work, 
and send problems to John.Cremona@nottingham.ac.uk.

Version compiled on Feb 11 2003 at 17:40:15 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floating point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,0,0,-36,0]

Curve [0,0,0,-36,0] :     
3 points of order 2:
[0 : 0 : 1], [6 : 0 : 1], [-6 : 0 : 1]

****************************
* Using 2-isogeny number 1 *
****************************

Using 2-isogenous curve [0,0,0,144,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 1
rk(S^{phi}(E'))=     3
rk(S^{phi'}(E))=       0

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 1
rk(phi'(S^{2}(E)))=  3
rk(phi(S^{2}(E')))=    0
rk(S^{2}(E))=  3
rk(S^{2}(E'))= 2

****************************
* Using 2-isogeny number 2 *
****************************

Using 2-isogenous curve [0,-36,0,36,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 1
rk(S^{phi}(E'))=     3
rk(S^{phi'}(E))=       0

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 1
rk(phi'(S^{2}(E)))=  3
rk(phi(S^{2}(E')))=    0
rk(S^{2}(E))=  3
rk(S^{2}(E'))= 2

****************************
* Using 2-isogeny number 3 *
****************************

Using 2-isogenous curve [0,36,0,36,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 1
rk(S^{phi}(E'))=      2
rk(S^{phi'}(E))=       1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 1
rk(phi'(S^{2}(E)))=  2
rk(phi(S^{2}(E')))=    1
rk(S^{2}(E))=  3
rk(S^{2}(E'))= 2

After second local descent, combined upper bound on rank = 1
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-18,72)
(c',d')=(36,36)
First stage (no second descent yet)...
(3,0,-18,0,24):  (x:y:z) = (1:3:1)
        Curve E         Point [3 : 9 : 1], height = 0.888625874839619
After first global descent, this component of the rank = 2
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(-2,0,36,0,-18):  (x:y:z) = (1:4:1)
       Curve E'        Point [-2 : -8 : 1], height = 0.44431293741981
  Curve E         Point [4 : 8 : 1], height = 0.888625874839619
After first global descent, this component of the rank = 1

-------------------------------------------------------
Summary of results:
-------------------------------------------------------
        rank(E) = 1
     #E(Q)/2E(Q) = 8

Information on III(E/Q):
 #III(E/Q)[phi']    = 1
  #III(E/Q)[2]       = 1

Information on III(E'/Q):
 #phi'(III(E/Q)[2]) = 1
  #III(E'/Q)[phi]    = 1
  #III(E'/Q)[2]      = 1

-------------------------------------------------------

List of points on E = [0,0,0,-36,0]:

I.  Points on E mod phi(E')
Point [-3 : 9 : 1], height = 0.888625874839619

II. Points on phi(E') mod 2E
Point [-2 : 8 : 1], height = 0.888625874839619
-------------------------------------------------------
Computing full set of 2 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.

Rank = 1
After descent, rank of points found is 1

Generator 1 is [-3 : 9 : 1]; height 0.888625874839619

The rank has been determined unconditionally.
The basis given is for a subgroup of full rank of the Mordell-Weil group
 (modulo torsion), possibly of index greater than 1.
Regulator (of this subgroup) = 0.888625874839619

 (20.2 seconds)
Enter curve: [0,0,0,0,0]

bash-2.05a$