2006 is a Congruent Number

[2006.02.04]合同数 2006

• 有理直角三角形を計算するプログラム(pari/GP)

$ mwrank3 -b 14 -c 14
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 Mar 22 2003 at 13:23:23 by GCC 3.2.1
using base arithmetic option LiDIA_ALL (LiDIA bigints and multiprecision floatin
g point)
Using LiDIA multiprecision floating point with 15 decimal places.
Enter curve: [0,0,0,-4024036,0]

Curve [0,0,0,-4024036,0] :
3 points of order 2:
[0 : 0 : 1], [2006 : 0 : 1], [-2006 : 0 : 1]

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

Using 2-isogenous curve [0,0,0,16096144,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        4
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     4
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

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

Using 2-isogenous curve [0,-12036,0,4024036,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        4
rk(S^{phi'}(E))=        1

-------------------------------------------------------
Second step, determining 2nd descent Selmer groups
-------------------------------------------------------
After second local descent, rank bound = 3
rk(phi'(S^{2}(E)))=     4
rk(phi(S^{2}(E')))=     1
rk(S^{2}(E))=   5
rk(S^{2}(E'))=  4

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

Using 2-isogenous curve [0,12036,0,4024036,0]
-------------------------------------------------------
First step, determining 1st descent Selmer groups
-------------------------------------------------------
After first local descent, rank bound = 3
rk(S^{phi}(E'))=        3
rk(S^{phi'}(E))=        2

-------------------------------------------------------
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))=   5
rk(S^{2}(E'))=  3

After second local descent, combined upper bound on rank = 2
Third step, determining E(Q)/phi(E'(Q)) and E'(Q)/phi'(E(Q))
-------------------------------------------------------
1. E(Q)/phi(E'(Q))
-------------------------------------------------------
(c,d)  =(-6018,8048072)
(c',d')=(12036,4024036)
First stage (no second descent yet)...
(17,0,-6018,0,473416):  (x:y:z) = (10:204:1)
        Curve E         Point [1700 : 34680 : 1], height = 3.79708469113873
(59,0,-6018,0,136408):  (x:y:z) = (3:295:1)
        Curve E         Point [531 : 52215 : 1], height = 3.79708469113873
After first global descent, this component of the rank = 3
-------------------------------------------------------
2. E'(Q)/phi'(E(Q))
-------------------------------------------------------
First stage (no second descent yet)...
(17,0,12036,0,236708):  no rational point found (hlim=8)
After first global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)
Second descent will attempt to reduce this
Second stage (using second descent)...
d1=17:
Second descent inconclusive for d1=17: ELS descendents exist but no rational poi
nt found

After second global descent, this component of the rank
        has lower bound 0
        and upper bound 1
        (difference =   1)

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

Information on III(E/Q):
        #III(E/Q)[phi']    <= 4
        #III(E/Q)[2]       is between 2 and 4

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

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

List of points on E = [0,0,0,-4024036,0]:

I.  Points on E mod phi(E')
Point [-306 : 34680 : 1], height = 3.79708469113873
Point [-1475 : 52215 : 1], height = 3.79708469113873

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

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

1 <= rank <= 2
Height Constant = 7.95047155880185

Max height = 3.79708469113873
Bound on naive height of extra generators = 8.37236985781727
After point search, rank of points found is 1

Generator 1 is [-306 : 34680 : 1]; height 3.79708469113873

The rank has not been completely determined,
only a lower bound of 1 and an upper bound of 2.

If the lower bound is strict, the basis given is for the full Mordell-Weil group
 (modulo torsion).
Regulator (of this subgroup) = 3.79708469113873
 (4.42e+04 seconds)
Enter curve: [0,0,0,0,0]

gp>  e=ellinit([0,0,0,-2006^2,0])
time = 102 ms.
%1 = [0, 0, 0, -4024036, 0, 0, -8048072, 0, -16192865729296, 193153728, 0, 4170283176822614953984, 1728, [2006.000000000000000000000000, 0.E-28, -2006.000000000000000000000000]~, 0.05854324022534062464932538198, 0.05854324022534062464932538198*I, -26.83138686462681503361636868, -80.49416059388044510084910605*I, 0.003427310976081940591969140468]
gp>  ellglobalred(e)
time = 55 ms.
%2 = [64384576, [1, 0, 0, 0], 64]
gp>  elllseries(e,1.00001)
time = 1mn, 14,427 ms.
%3 = 0.00007112928828965770414582745048
gp>  elllseries(e,2-1.00001)
time = 1mn, 21,032 ms.
%4 = -0.00007113864254779321396001499387
gp>  ellrootno(e,1)
time = 11 ms.
%5 = -1

gp>  read("./congr.gp")
time = 59 ms.
gp>  ss(e,[-306,34680],2006,10)
Pi/2-congruent number 2006
2([-306, 34680]): [177/5, 340/3, 1781/15]
4([-306, 34680]): [2608039/53430, 214361160/2608039, 13320805745521/139347523770]
6([-306, 34680]): [1096292682768580/89598005203437, 5286282307002783/16121951217185, 473970003717484279677323566421/1444494669046899105695464845]
8([-306, 34680]): [84913064761177931647318559/3712442590519080237068340, 14894319673162549911118180080/84913064761177931647318559, 55762422389013803519428353258482450280167593410156481/315234878110901825802045825733466627868996233322060]
10([-306, 34680]): [1060906838709427440516351853707632258006300/17057675849220703606218823672724068607283, 1006402875104021512766910596690720047829697/15601571157491580007593409613347533205975, 23846693303611761611740400493200164768822262512210855596176575257704558637246164501/266126543543042422733515318496424223840561094670028557337389273736113914224115925]
time = 110 ms.
gp>  (177/5)^2+(340/3)^2-(1781/15)^2
time = 0 ms.
%6 = 0
gp>  (1/2)*(177/5)*(340/3)
time = 0 ms.
%7 = 2006
gp>  (2608039/53430)^2+(214361160/2608039)^2-(13320805745521/139347523770)^2
time = 0 ms.
%8 = 0
gp>  (1/2)*(2608039/53430)*(214361160/2608039)
time = 0 ms.
%9 = 2006

[参考文献]

• [1]Karl Rubin, "Right triangles and elliptic curves", p1-8.
  
• [2]Bjorn Poonen, "Elliptic Curves", March 2001, p8.
  

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