Zhengguoping on Thu, 30 Nov 2000 16:27:47 +0800

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 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=1.635634785753706484129322088w=1 g=-0.02032860177457994183799764121N=[2, 1; 13, 1; 19, 1]? sha(80,80)   A=80, B=80, sha=2.024898913205420105333092233w=1 g=-0.05536173020696967026317163944N=[5, 2; 347, 1]? sha(93,94)   A=93, B=94, sha=1.601903389510516187331764338w=1 g=-0.001695775959811050535985916630N=[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

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