A question

Here is a question. I use GP version 2.0.20 (Windows 98/Dos) to do some computations concerning elliptic curves. According to BSD conjecture, if L(E, 1) is not zero, then $shafa =L(E,1)*E_tors^2/(omega*C)$. Since shafa is a square, sqrt(shafa) should be an integer. I verify this for elliptic curves $y^2=x^3+Ax+B$ for A from 61 to 100, B from 61 to 100. All are OK except three cases. See follows.

***************************

\\ Here is the script.

\\ \l
\\ red;
\\ Em;
sha(A,B,E,delta,w,Omega,C,L1,L2,Etors,shafa,red,Em)=
{
delta=-16*(4*A^3+27*B^2);
E=[0,0,0,A,B];
if(delta!=0,
Em=ellinit(E);
w=ellrootno(Em);
Omega=Em.omega[1];
red=ellglobalred(Em);
C=red[3];
L1=elllseries(Em, 1);
L2=abs(elllseries(Em,1.0000000000001,1));
Etors=elltors(Em)[1];
shafa=sqrt(L1*Etors^2/(C*Omega)), shafa=-1);
\\ shafa=L1*Etors^2/(C*Omega), shafa=-1);
print();
print("A=",A,","," B=",B,"," ," sha=",shafa);
if(L2!=0.0,
print("w=", w , " g=",log(L2)/log(0.0000000000001)), );
print("N=", factor(red[1]));
\\ write(sha.log, shafa);
}
/*
for(A=61,100, for(B=61, 100,
print("A=",A,","," B=",B,"," ,"sha=",sha(A,B),",", "w=", w);
print("g=",log(elllseries(Em,1.0000001))/log(0.0000001));
print("N=", factor(red[1]))
));*/

sha(61,62)
sha(80,80)
sha(93,94)

\\ for(A=61,100, for(B=61, 100, sha(A,B)))
\\ \l

**************************

here is the result.

? \rsha.gp
? sha(A,B,E,delta,w,Omega,C,L1,L2,Etors,shafa,red,Em)=delta=-16*(4*A^3+27*B^2);E=[0,0,0,A,B];if(delta!=0,Em=ellinit(E);w=ellrootno(Em);Omega=Em.omega[1];red=ellglobalred(Em);C=red[3];L1=elllseries(Em,1);L2=abs(elllseries(Em,1.0000000000001,1));Etors=elltors(Em)[1];shafa=sqrt(L1*Etors^2/(C*Omega)),shafa=-1);print();print("A=",A,","," B=",B,","," sha=",shafa);if(L2!=0.0,print("w=",w," g=",log(L2)/log(0.0000000000001)),);print("N=",factor(red[1]));
? sha(61,62)

A=61, B=62, sha=1.635634785753706484129322088
w=1 g=-0.02032860177457994183799764121
N=[2, 1; 13, 1; 19, 1]
? sha(80,80)

A=80, B=80, sha=2.024898913205420105333092233
w=1 g=-0.05536173020696967026317163944
N=[5, 2; 347, 1]
? sha(93,94)

A=93, B=94, sha=1.601903389510516187331764338
w=1 g=-0.001695775959811050535985916630
N=[2, 1; 3, 2; 5, 1]
? \l
log = 0 (off)
[logfile was "sha2.log"]

***************************************************

I run the script on an intel 586/Windows98.

Please check it. Could you tell me why?

Sincerely yours,

G.P.Zheng from Nanjing, China. 00-11-27.

My email address: gzheng@jlonline.com