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,-228,848]

Curve [0,0,0,-228,848] :     
1 points of order 2:
[4 : 0 : 1]

Using 2-isogenous curve [0,-24,0,864,0]
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First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 2
rk(S^{phi}(E'))=  3
rk(S^{phi'}(E))=       1

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

Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(12,-180)
(c',d')=(-24,864)
First stage (no second descent yet)...
(-2,0,12,0,90):  (x:y:z) = (1:10:1)
        Curve E         Point [-2 : -20 : 1], height = 1.64739216431184
(3,0,12,0,-60):  (x:y:z) = (2:6:1)
       Curve E         Point [12 : 36 : 1], height = 0.906408610203134
After first global descent, this component of the rank = 3
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
This component of the rank is 0

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Summary of results:
-------------------------------------------------------
      rank(E) = 2
     #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,-228,848]:

I.  Points on E mod phi(E')
Point [2 : -20 : 1], height = 1.64739216431184
Point [16 : 36 : 1], height = 0.906408610203134

II.  Points on phi(E') mod 2E
--none (modulo torsion).

-------------------------------------------------------
Computing full set of 4 coset representatives for
2E(Q) in E(Q) (modulo torsion), and sorting into height order....done.

Rank = 2
After descent, rank of points found is 2

Generator 1 is [2 : -20 : 1]; height 1.64739216431184
Generator 2 is [16 : 36 : 1]; height 0.906408610203134

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) = 0.358602402413191

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

bash-2.05a$