[was@laptop 02-25-03]$ meccah was@meccah.math.harvard.edu's password: Last login: Tue Feb 25 15:49:01 2003 from roam240-72.fas.harvard.edu ********************************************************************** Welcome to MECCAH.MATH.HARVARD.EDU Mathematics Extreme Computation Cluster At Harvard 12 Athlon 2000 MP processors (2 are only 1900) master node has 3 GB memory and the other five have 2GB each openMosix 2.4.19 with RedHat Linux 7.3 and Kernel 2.4.19 Administrator: William Stein (was@math.harvard.edu) HINTS: * Type mosmon and mtop for CPU usage and multi-process top. * Type ulimit -S -v 2000000 etc to use about 2G memory per process, instead of the default 900MB. 09/29/02: Upgraded to Redhat 7.3, Kernel 2.4.19, openMosix 2.4.19. 09/29/02: Rebooted because of "slapper worm". 10/14/02: Rebooted because power off (due to construction). 10/23/02: I ran a bunch of HUGE memory jobs, which resulted in a mosix "VM memory overcommit" error message, and a reboot. 11/14/02: MECCAH3 is back from the shop, and works so far... 11/16/02: Changed gp back to 2.1.4 (from 2.2.alpha) 11/26/02: MECCAH1 crashed when we found a bug in openMosix. 01/19/03: MECCAH3 crashed. MECCAH1 was overloaded probably, since it responded to pings but not much else. 01/25/03: I installed Macaulay 2-0.9.2. 01/25/03: MECCAH3 has stopped working again. I'll fix it myself this time, so it will really get fixed. 02/07/03: Rebooted, since things were locked up because of meccah3. ********************************************************************** [was@meccah was]$ mwrank 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 Dec 31 2000 at 15:35:35 by GCC egcs-2.91.66 19990314/Linux (egcs-1.1.2 release) using base arithmetic option NTL (NTL bigints, no multiprecision floating point and no nontrivial integer factorization) Enter curve: [0,0,0,0,1] Curve [0,0,0,0,1] : 1 points of order 2: [-1 : 0 : 1] Using 2-isogenous curve [0,0,0,-15,22] ------------------------------------------------------- First step, determining Selmer group ------------------------------------------------------- ------------------------------------------------------- Rank = 0 ------------------------------------------------------- Second step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q)) ------------------------------------------------------- 1. E(Q)/phi(E'(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- 2. E'(Q)/phi'(E(Q)) ------------------------------------------------------- This component of the rank is 0 ------------------------------------------------------- Summary of results: ------------------------------------------------------- rank(E) = 0 #E(Q)/2E(Q) = 2 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 Rank = 0 After descent, rank of points found is 0 The rank and full Mordell-Weil basis have been determined unconditionally. Regulator = 1 (0.00 seconds) Enter curve: [1,2,3,4,5] Curve [1,2,3,4,5] : Working with minimal curve [1,-1,0,4,3] [u,r,s,t] = [1,-1,0,-1] No points of order 2 Basic pair: I=-183, J=-6858 disc=-71546112 2-adic index bound = 2 By Lemma 5.1(b), 2-adic index = 1 2-adic index = 1 One (I,J) pair *** BSD give two (I,J) pairs Looking for quartics with I = -183, J = -6858 Looking for Type 3 quartics: Trying positive a from 1 up to 6 (1,2,-9,22,-11) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [2 : 3 : 1] height = 1.26539379176449 Rank of B=im(eps) increases to 1 (4,3,6,3,-4) --trivial Trying negative a from -1 down to -5 (-4,3,6,3,4) --trivial Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [1 : 2 : 1], height = 1.26539379176449 After descent, rank of points found is 1 Transferring points back to original curve [1,2,3,4,5] Generator 1 is [1 : 2 : 1]; height 1.26539379176449 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) = 1.26539379176449 (0.00 seconds) Enter curve: [3,8,2,5,6] Curve [3,8,2,5,6] : Working with minimal curve [1,1,1,-27,48] [u,r,s,t] = [1,-3,-1,4] No points of order 2 Basic pair: I=1297, J=-102706 disc=-1821222144 2-adic index bound = 2 By Lemma 5.1(b), 2-adic index = 1 2-adic index = 1 One (I,J) pair *** BSD give two (I,J) pairs Looking for quartics with I = 1297, J = -102706 Looking for Type 3 quartics: Trying positive a from 1 up to 13 (1,2,23,106,117) --nontrivial...(x:y:z) = (1 : 1 : 0) Point = [-4 : 12 : 1] height = 2.04216869613643 Rank of B=im(eps) increases to 1 (6,3,-10,11,18) --trivial Trying negative a from -1 down to -2 Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 1 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 0 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Rank = 1 Points generating E(Q)/2E(Q): Point [-7 : 20 : 1], height = 2.04216869613643 After descent, rank of points found is 1 Transferring points back to original curve [3,8,2,5,6] Generator 1 is [-7 : 20 : 1]; height 2.04216869613643 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) = 2.04216869613643 (0.01 seconds) Enter curve: [0,0,0,0,7823] Curve [0,0,0,0,7823] : No points of order 2 Basic pair: I=0, J=-211221 disc=-44614310841 2-adic index bound = 2 2-adic index = 2 Two (I,J) pairs Looking for quartics with I = 0, J = -211221 Looking for Type 3 quartics: Trying positive a from 1 up to 17 Trying negative a from -1 down to -11 Finished looking for Type 3 quartics. Looking for quartics with I = 0, J = -13518144 Looking for Type 3 quartics: Trying positive a from 1 up to 68 (30,-12,48,116,-18) --nontrivial...locally soluble...no rational point found (limit 10) --new (B) #1 (41,-16,-6,112,-11) --nontrivial...--equivalent to (B) #1 Trying negative a from -1 down to -45 (-11,-20,408,1784,2072) --nontrivial...--equivalent to (B) #1 (-18,-28,312,996,838) --nontrivial...--equivalent to (B) #1 Finished looking for Type 3 quartics. Mordell rank contribution from B=im(eps) = 0 Selmer rank contribution from B=im(eps) = 1 Sha rank contribution from B=im(eps) = 1 Mordell rank contribution from A=ker(eps) = 0 Selmer rank contribution from A=ker(eps) = 0 Sha rank contribution from A=ker(eps) = 0 Warning: Selmer rank = 1 and program finds lower bound for rank = 0 which differs by an odd integer from the Selmer rank. Hence the rank must be 1 more than reported here. Try rerunning with a higher bound for quartic point search. Summary of results (all should be powers of 2): n0 = #E(Q)[2] = 1 n1 = #E(Q)/2E(Q) >= 1 n2 = #S^(2)(E/Q) = 2 #III(E/Q)[2] <= 2 0 <= rank <= selmer-rank = 1 0 <= rank <= selmer-rank = 1 After descent, rank of points found is 0 The rank has not been completely determined, only a lower bound of 0 and an upper bound of 1. Standard parity conjectures would increase the lower bound by 1 to 1, implying that the rank was exactly 1. (0.17 seconds) Enter curve: